Hif 12018
Hif 12018
Hif 12018
FHWA-HIF-12-018
April 2012
U.S. Department
of Transportation
Federal Highway
Administration
1.
4.
2.
3.
5.
Report Date
7.
April 2012
Author(s)
6.
8.
10.
11.
DTFH61-06-D-00010
12.
15.
13.
14.
Supplementary Notes
Technical Project Manager: Dr. L.A. Arneson (FHWA Resource Center Hydraulics)
Technical Assistance: Joe Krolak (FHWA Headquarters Hydraulics) and Dr. Kornel Kerenyi (FHWA Research Hydraulics)
16.
This document provides technical information and guidance on the hydraulic analysis and design of bridges. The goal is to
provide information such that bridges can be designed as safely as possible while optimizing costs and limiting impacts to
property and the environment. Many significant aspects of bridge hydraulic design are discussed. These include regulatory
topics, specific approaches for bridge hydraulic modeling, hydraulic model selection, bridge design impacts on scour and
stream instability, and sediment transport.
17.
Key Words
18.
20.
Distribution Statement
This document is available to the public through the
National Technical Information Service,
Springfield, VA 22161 (703) 487-4650
21.
No. of Pages
280
22.
Price
TABLE OF CONTENTS
LIST OF FIGURES ................................................................................................................ vii
LIST OF TABLES ..................................................................................................................xiii
LIST OF SYMBOLS .............................................................................................................. xv
LIST OF ACRONYMS .......................................................................................................... xix
GLOSSARY ......................................................................................................................... xxi
ACKNOWLEDGMENTS ................................................................................................... xxxvii
CHAPTER 1 - INTRODUCTION .......................................................................................... 1.1
1.1
1.1.1
1.1.2
1.1.3
1.2
1.3
1.3.1
1.3.2
1.3.3
1.3.4
1.3.5
1.3.6
1.3.7
1.3.8
3.2
3.2.1
3.2.2
3.2.3
3.3
3.3.1
3.3.2
3.3.3
3.3.4
3.4
3.4.1
3.4.2
3.4.3
3.4.4
ii
5.4
5.4.1
5.4.2
5.5
5.5.1
5.5.2
5.5.3
5.6
5.6.1
5.6.2
5.6.3
5.7
5.7.1
5.7.2
5.7.3
5.7.4
5.8
5.8.1
5.8.2
5.8.3
5.8.4
5.8.5
5.9
5.9.1
5.9.2
5.9.3
5.9.4
5.9.5
6.5
6.5.1
6.5.2
6.5.3
6.6
6.7
6.8
6.9
6.9.1
6.9.2
6.9.3
6.9.4
6.10
6.10.1
6.10.2
6.10.3
6.10.4
iv
8.4
8.4.1
8.4.2
8.5
8.5.1
8.5.2
8.5.3
9.3.1
9.3.2
9.3.3
9.4
9.5
9.6
10.5
10.5.1
10.5.2
10.5.3
10.5.4
vi
LIST OF FIGURES
Figure 3.1. Flow in the X-Z plane and flow in terms of streamline and normal
coordinates ....................................................................................................... 3.2
Figure 3.2. Example of uniform flow (Y2 = Y1) .................................................................... 3.3
Figure 3.3. Example of nonuniform flow where the depth of flow Y2 Y1 .......................... 3.4
Figure 3.4. Streamtube with fluid flowing from Section 1 to Section 2 ............................... 3.5
Figure 3.5. Net flow through a control volume.................................................................... 3.7
Figure 3.6. Surfaces forces acting on a fluid element in the X and Y directions for
an inviscid fluid ............................................................................................... 3.8
Figure 3.7. Nonuniform velocity distribution ..................................................................... 3.10
Figure 3.8. The control volume for conservation of linear momentum ............................. 3.11
Figure 3.9. Forces acting on a control volume for uniform flow conditions ...................... 3.13
Figure 3.10. Floodplain roughness example ...................................................................... 3.19
Figure 3.11. Looking upstream from the right bank on Indian Fork, near
New Cumberland, Ohio ................................................................................. 3.20
Figure 3.12. An example of debris blockage on piers ....................................................... 3.22
Figure 3.13. An example of upstream bridge cross section with debris
accumulation on a single pier ........................................................................ 3.22
Figure 3.14. Weir flow over a sharp crested weir .............................................................. 3.23
Figure 3.15. Example of simplified weir flow for a sharp crested weir............................... 3.24
Figure 3.16. Ogee spillway crest ....................................................................................... 3.25
Figure 3.17. Discharge coefficients for vertical-face ogee crest ....................................... 3.26
Figure 3.18. Ratio of discharge coefficients caused by tailwater effects ........................... 3.26
Figure 3.19. Broad-crested weir ........................................................................................ 3.27
Figure 3.20. Discharge reduction factor versus percent of submergence ......................... 3.28
Figure 3.21. Sluice and tainter gates................................................................................. 3.29
Figure 3.22. Flow classification according to change in depth with respect to
space and time .............................................................................................. 3.31
vii
ix
Figure 7.12. Illustration of an off-line storage area using the HEC-RAS computed
model............................................................................................................ 7.15
Figure 7.13. Cross section hydraulic table increments...................................................... 7.16
Figure 7.14. Conveyance properties versus elevation for a single cross section.............. 7.17
Figure 7.15. Channel cross section for an unsteady flow model ....................................... 7.18
Figure 7.16. Conveyance properties versus elevation for a single cross section
with right overbank subdivided ...................................................................... 7.19
Figure 8.1. Velocity and streamlines at a bridge constriction ............................................. 8.2
Figure 8.2. Live-bed contraction scour variables................................................................ 8.5
Figure 8.3. Clear-water contraction scour variables ........................................................... 8.6
Figure 8.4. Vertical contraction scour ................................................................................. 8.7
Figure 8.5. The main flow features forming the flow field at a cylindrical pier .................... 8.8
Figure 8.6. Flow structure in floodplain and main channel at a bridge opening ................. 8.9
Figure 8.7. View down at debris and scour hole at upstream end of pier ........................ 8.10
Figure 8.8. Idealized flow pattern and scour at pier with debris ....................................... 8.11
Figure 8.9. Debris in HEC-RAS hydraulic model.............................................................. 8.11
Figure 8.10. Headcut downstream of a bridge ................................................................... 8.13
Figure 8.11. Meander migration on Wapsipinicon River near De Witt, Iowa ...................... 8.13
Figure 8.12. Channel widening and meander migration on Carson River near
Weeks, Nevada ............................................................................................. 8.14
Figure 8.13. Cross section locations at bridge crossings in one-dimensional models ....... 8.15
Figure 8.14. Flow distribution from one-dimensional models ............................................. 8.16
Figure 8.15. Velocity and flow lines in two-dimensional models ........................................ 8.17
Figure 8.16. Typical guide bank ......................................................................................... 8.18
Figure 8.17. Guide bank in a two-dimensional network ..................................................... 8.19
Figure 8.18. Flow field at a bridge opening with guide bank .............................................. 8.20
Figure 8.19. Two-dimensional analysis of flow along spurs, (a) flow field without
spurs, and (b) flow field with spurs ................................................................ 8.21
xi
Figure 9.1. Definition sketch of the sediment continuity concept ....................................... 9.3
Figure 9.2. Definitions of sediment load components ....................................................... 9.5
Figure 9.3. Suspended sediment concentration profiles ................................................... 9.6
Figure 9.4. Bed forms in sand channels ............................................................................ 9.8
Figure 9.5. Relative resistance to flow in sand-bed channels ........................................... 9.9
Figure 9.6. Velocity and sediment concentration profiles ................................................. 9.11
Figure 9.7. Channel profiles from sediment routing model............................................... 9.13
Figure 9.8. Contraction scour and water surface for fixed-bed and mobile-bed
models ............................................................................................................ 9.14
Figure 10.1. CFD results plot showing velocity direction and magnitude from a
model of a six-girder bridge .......................................................................... 10.2
Figure 10.2. Definition sketch for deck force variables..................................................... 10.3
Figure 10.3. Drag coefficient for 6-girder bridge............................................................... 10.4
Figure 10.4. Lift coefficient for 6-girder bridge.................................................................. 10.4
Figure 10.5. Moment coefficient for 6-girder bridge ......................................................... 10.5
Figure 10.6. Photograph of a bridge damaged by Hurricane Katrina from HEC-25 ......... 10.6
Figure 10.7. Illustration of a confluence situation ........................................................... 10.10
Figure 10.8. Velocity from physical modeling using Particle Image Velocimetry ........... 10.12
Figure 10.9. Velocity result from CFD (RANS) modeling ............................................... 10.12
Figure 10.10. Hybrid modeling of pier scour .................................................................... 10.13
Figure 10.11. Illustration of CFD modeling of Woodrow Wilson Bridge pier
and dolphins ............................................................................................... 10.14
Figure 10.12. Photograph of the post-scour condition of a small-scale physical
model test of a Woodrow Wilson Bridge pier ............................................. 10.15
Figure 10.13. Sketch illustrating spread width of bridge deck drainage ........................... 10.17
Figure 10.14. Installed underdeck bridge drainage system.............................................. 10.19
xii
LIST OF TABLES
Table 1.1. Commonly Used Engineering Terms in U.S. Customary and SI Units............... 1.6
Table 3.1. Values for the Computation of the Manning Roughness Coefficient
using Equation 3.43......................................................................................... 3.17
Table 3.2. Values of the Manning Roughness Coefficient for Natural Channels .............. 3.18
Table 3.3. Typical Drag Coefficients for Different Pier Shapes ......................................... 3.21
Table 3.4. Summary of the Flow Profiles .......................................................................... 3.45
Table 4.1. Bridge Hydraulic Modeling Selection ................................................................. 4.8
Table 4.2. Data Used in Bridge Hydraulic Studies ............................................................ 4.17
Table 5.1. Ranges of Expansion Rates, ER ...................................................................... 5.16
Table 5.2. Ranges of Contraction Rates, CR .................................................................... 5.17
Table 5.3. Yarnell Pier Shape Coefficients........................................................................ 5.24
xiii
xiv
LIST OF SYMBOLS
a
a
aproj
A
AE
AP
An
c
c
ca
C
C
b
bproj
B
Cc
Cd
CD
CL
Cm
CR
Cr
CW
D
Dm
Ds
D50
e
E
EGL
ER
f
F
FD
Ff
FM
FP
Fr
g
h
hb
hf
hu
h1*
he
hL
h*
H
HGL
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
K,K ij
K
K*
ks
Ku
L
L
M, m
M
Mcg
m
n
nc
ne
nob
n0,1,2,3,4
n
P, p
q
qb
ql
Q
qs
Qs
Qw
r
R
Re
s
s
S
S0
Sc
Se
Sf
Sw
t
t
T
U
v
V
V
V
Vc
Vc
Vn
v*
W
W
WP
WS
WSEL
Wx
X, x
Y, y
Y, y
y0
yc
yh
ys
ys-a
ys-c
ys-p
ys-vc
Y
YB
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
YP
Z, z
z
w
s
0
b
c
s
xx, xy
yy, yx
xviii
LIST OF ACRONYMS
AASHTO
AREMA
ASCE
BEM
CADD
CEM
CFD
CFR
CLOMR
CWA
DOT
EGL
EPA
FDM
FEM
FEMA
FESWMS
FHWA
FIS
FSA
FST2DH
GIS
HDS
HEC
HEC
HGL
LES
LIDAR
LRFD
NCHRP
NED
NEPA
NOAA
NOS
NRC
NRCS
OHW
PIV
RANS
RAS
RMA2
SBR
SCS
SI
SIAM
SMS
TAC
TRACC
TRB
UNET
USACE
USBR
USCG
USDA
USDOT
USGS
WSEL
WSPRO
xx
GLOSSARY
adverse slope:
The hydraulic condition where the bed slope in the direction of flow
is negative and normal depth is undefined.
aggradation:
alluvial channel:
alluvial fan:
alluvial stream:
alluvium:
annual flood:
approach section:
average velocity:
avulsion:
backwater:
bank:
bank protection:
xxi
bank revetment:
bankfull discharge:
base floodplain:
bathymetry:
bed:
bed layer:
bed load:
bed material:
bedrock:
bed shear
(tractive force):
bed slope:
boulder:
boundary condition:
bridge opening:
bridge owner:
bridge section:
bridge waterway:
bulk density:
causeway:
caving:
channel:
channel pattern:
channelization:
check dam:
clay (mineral):
clear-water scour:
cobble:
Computational Fluid
Dynamics (CFD):
confluence:
constriction:
contact load:
contraction reach:
The river reach were flow is converging from being fully expanded
in the floodplain into the bridge opening.
xxiii
contraction scour:
contraction:
control section:
conveyance:
Coriolis force:
countermeasure:
critical depth:
critical slope:
critical velocity:
cross section:
crossing:
current:
cut bank:
cutoff:
debris:
degradation (bed):
depth of scour:
dike:
discharge:
drag force:
drift:
eddy current:
energy correction
coefficient ():
ephemeral stream:
Stream or reach of stream that does not flow for parts of the year.
As used here, the term includes intermittent streams with flow less
than perennial.
equilibrium scour:
Scour depth in sand-bed stream with dune bed about which live
bed pier scour level fluctuates due to variability in bed material
transport in the approach flow.
xxv
erosion:
exit section:
expansion reach:
The river reach were flow is diverging from the bridge opening until
it is fully expanded into the floodplain.
extent:
fall velocity:
fetch:
fetch length:
Horizontal distance (in the direction of the wind) over which wind
generates waves and wind setup.
fill slope:
flashy stream:
flood-frequency curve:
Graph indicating the probability that the annual flood discharge will
exceed a given magnitude, or the recurrence interval
corresponding to a given magnitude.
floodplain:
flow hazard:
flow profile:
flow resistance:
fluvial geomorphology:
fluvial system:
freeboard:
Froude Number:
geomorphology/
morphology:
That science that deals with the form of the Earth, the general
configuration of its surface, and the changes that take place due to
erosion and deposition.
grade-control structure
(sill, check dam):
graded stream:
gravel:
guide bank:
headcutting:
helical flow:
horizontal slope:
The hydraulic conditions where the bed slope is zero and normal
depth is infinite.
hydraulic control:
hydraulic model:
hydraulic radius:
hydraulic structures:
hydraulics:
hydrograph:
hydrology:
hydrostatic pressure:
incipient overtopping:
The condition when flow is at the road crest, but not flowing over
the road.
ineffective flow:
initiation of motion:
The hydraulic condition when bed material, often the median grain
size, begins to move and sediment transport of bed material
occurs.
invert:
island:
lateral erosion:
levee:
live-bed scour:
local scour:
longitudinal profile:
lower bank:
mathematical model:
meander or full
meander:
meandering stream:
median diameter:
migration:
mild slope:
The hydraulic condition where the bed slope is less than critical
slope and normal depth is greater than critical depth.
model extent:
xxix
momentum correction
coefficient ():
mud:
multiple openings:
natural levee:
Low ridge that slopes gently away from the channel banks that is
formed along streambanks during floods by deposition.
non-uniform flow:
normal depth:
A condition when the water surface slope and energy grade slope
are parallel to the bed slope. Also a boundary condition where the
water surface is computed from a preset energy grade slope.
normal stage:
numerical model:
one-dimensional model:
orifice flow:
overbank flow:
Water movement that overtops the bank either due to stream stage
or to overland surface water runoff.
overtopping flow:
parallel bridges:
Bridges located in series along a channel where flow does not fully
expand between the bridges.
perennial stream:
phreatic line:
physical model:
xxx
pile:
probable maximum
flood:
rapid drawdown:
Lowering of the water against a bank more quickly than the bank
can drain without becoming unstable.
reach:
recurrence interval:
regime:
relief bridge:
revetment:
riparian:
riprap:
river training:
roughness coefficient:
runoff:
saltation load:
sand:
scour:
sediment concentration:
sediment continuity:
sediment discharge:
sediment load:
sediment or fluvial
sediment:
sediment yield:
seepage:
shear stress:
silt:
similitude:
sinuosity:
xxxii
skew:
slope-area method:
sloughing:
slump:
specific energy:
spill-through abutment:
spur dike:
spur:
stability:
stable channel:
Condition that exists when a stream has a bed slope and cross
section which allows its channel to transport the water and
sediment delivered from the upstream watershed without
aggradation, degradation, or bank erosion (a graded stream).
stage:
steady flow :
steep slope:
The hydraulic condition where the bed slope is greater than critical
slope and normal depth is less than critical depth.
xxxiii
stream:
Body of water that may range in size from a large river to a small
rill flowing in a channel. By extension, the term is sometimes
applied to a natural channel or drainage course formed by flowing
water whether it is occupied by water or not.
streambank erosion:
streambank failure:
streambank protection:
streamline:
streamtube:
subcritical, supercritical
flow:
Open channel flow conditions with Froude Number less than and
greater than unity, respectively.
Flow through a bridge where the upstream and downstream lowchords are submerged.
suspended sediment
discharge:
thalweg:
three-dimensional
model:
toe of bank:
topography:
total scour:
Sum of suspended load and bed load or the sum of bed material
load and wash load of a stream (total load).
xxxiv
tractive force:
turbulence:
two-dimensional model:
ultimate scour:
uniform flow:
unit discharge:
unsteady flow:
upper bank:
velocity:
vertical abutment:
vortex:
wash load:
watershed:
xxxvi
ACKNOWLEDGMENTS
This manual is a major revision and replacement of HDS 1 "Hydraulics of Bridge
Waterways," which was written by Joseph N. Bradley in 1960 and revised in 1970 and 1978.
The authors wish to acknowledge the contributions made by Mr. Bradley through his
landmark document.
DISCLAIMER
Mention of a manufacturer, registered or trade name does not constitute a guarantee or
warranty of the product by the U.S. Department of Transportation or the Federal Highway
Administration and does not imply their approval and/or endorsement to the exclusion of
other products and/or manufacturers that may also be suitable.
xxxvii
xxxviii
CHAPTER 1
INTRODUCTION HYDRAULIC DESIGN OF SAFE BRIDGES
1.1. INTRODUCTION
1.1.1 Background
The Federal Highway Administration provides oversight of the Nation's bridges through the
National Bridge Inspection Standards (NBIS) and other regulatory policies and programs.
Bridge failures resulting from both natural and human factors led the U.S. Congress to
express its concern about the safety, approaches, and oversight of the Nation's bridges.
Within the Conference Report for the Departments of Transportation and Housing and Urban
Development, and Related Agencies Appropriations Act, 2010 (H.R. Rep. No. 111-366), the
Congress recommended that the " (FHWA) use a more risk-based, data-driven approach
to its bridge oversight" to improve bridge safety. Congress stated its intention to monitor the
progress that FHWA makes in identifying new approaches to bridge oversight, completing
initiatives, and achieving results from its efforts. Congress directed that FHWA apply funds to
focus and achieve these activities.
To address the conference report, FHWA undertook a combination of activities that
contribute to four primary outcomes: more rigorous oversight of bridge safety; full NBIS
compliance by all States; improved information for safety oversight and condition monitoring;
and qualified and equipped bridge inspection personnel.
As hydraulic issues remain a leading factor in bridge failures, FHWA recognized that these
activities need to include efforts to better collect, understand and deploy more recent and
robust guidance and techniques to the accepted state of hydraulic and waterway related
practice. This document is one of the products of these efforts.
1.1.2 Purpose
The purpose of HDS 7, Hydraulic Design of Safe Bridges, is to provide technical information
and guidance on the hydraulic design of bridges. HDS 7 replaces the HDS 1 manual
"Hydraulics of Bridge Waterways" (FHWA 1978) for guidance of bridge hydraulic analyses.
Bridges should be designed as safely as possible while optimizing costs and limiting impacts
to property and the environment. Many significant aspects of bridge hydraulic design are
discussed. These include regulatory topics, specific approaches for bridge hydraulic
modeling, hydraulic model selection, bridge design impacts on scour and stream instability,
and sediment transport.
The impacts of bridge design and construction on the economics of highway design, safety to
the traveling public, and the natural environment can be significant. An economically viable
and safe bridge is one that is properly sized, designed, constructed, and maintained. In
general, although longer bridges are more expensive to design and build than shorter
bridges, they cause less backwater, experience less scour, and can reduce impacts to the
environment. Increased scour from too short a bridge can require deeper foundations and
necessitate countermeasures to resist these effects. A properly designed bridge is one that
balances the cost of the bridge with concerns of safety to the traveling public, impacts to the
environment, and regulatory requirements to not cause harm to those that live or work in the
floodplain upstream and downstream of the bridge.
1.1
V=
1.486 2 / 3 1 / 2
R S
n
(1.1)
where:
V = Velocity, ft/s
n = Roughness Coefficient
R = Hydraulic Radius, ft
S = Slope
There were two things that Robert Manning did not like about his equation, (1) that it was
dimensionally incorrect, and (2) it was difficult (at the time) to determine the cubed root of a
number and then square it to arrive at a number to the 2/3rd power. King's handbook (King
1918) presented a table of numbers from 0.01 to 10 to the 2/3rd power which eliminated the
problem of determining a number to the 2/3rd power and is considered to be a leading reason
in the early acceptance and of use of the Manning Equation.
1.2
As methods were being developed to estimate discharge, it was realized that one could
make an estimate of the roughness coefficient based on known values from similar channels
and floodplains, determine the slope of the channel, and then use an iterative solution to
determine the "normal" depth at a cross-section or hydraulic opening. Through the 1950s
this remained a popular method of determining the depth and velocity of flow at a crosssection or through a hydraulic opening.
The problem with using normal depth as the estimate of flow depth (and velocity) for
determining the size of hydraulic opening is that it does not consider the effects of backwater.
Backwater is the additional depth to accelerate flow through the bridge opening and
overcome a variety of resistance and drag forces. These forces depend on a number of
factors including bridge type, degree of contraction, embankment skew, pier number and
type, debris blockage, etc.
To account for backwater, research was completed and methods were developed that
examined the components of backwater (Liu et al. 1957). In HDS 1, the computed
backwater was added to the "normal" depth at a location upstream of the bridge to evaluate
the overall impacts of a bridge (FHWA 1978).
Another significant development that contributed to the development of the current state of
bridge hydraulics was the publication of a textbook about open channel flow by V.T. Chow
(Chow 1959). The publication presents and applies concepts of energy, momentum, and
continuity to the flow of water in open channels. It is also one of the places where the direct
and standard step methods for computing water surface profiles were first presented. The
direct step method is applicable to prismatic channels and the standard step method to
natural channels. The standard step method uses concepts of conservation of energy and
flow, and is widely used for water surface profile calculations.
At the same time the physics of open channel flow and water surface profiles were being
developed, mainframe computer and programming languages were developing. The
application of computer programs made it possible to rapidly complete trial and error
solutions required for computing water surface profiles. One of the first computer programs
that was developed to compute water surface profiles in natural channels was HEC-2
(USACE 1992) with development dating back to at least 1964. The HEC-2 program has
undergone continual development and was ported to the PC in 1984. HEC-2 has evolved
into the HEC-RAS (River Analysis System) model (USACE 2010a, b, c). HEC-RAS performs
steady non-uniform flow hydraulic calculations similar to HEC-2, but incorporates enhanced
visualization, more complete bridge and culvert hydraulic computations, unsteady flow, and
sediment transport. There were many other computer programs developed to compute
water surface profiles. The USGS developed E431 (USGS 1976) and the Federal Highway
Administration developed WSPRO (FHWA 1998) that had components specifically
formulated for the analysis of flow through bridge openings. HEC-RAS has incorporated
features from these programs including the WSPRO bridge routine.
More recent developments in the field of bridge hydraulics include the development of twodimensional hydraulic and hydrodynamic models to compute the entire flow field. These
models include FST2DH (FHWA 2003) and RMA-2 (USACE 2009).
1.2 HYDRAULIC ANALYSIS OVERVIEW
The hydraulic analysis of a bridge opening is a complicated undertaking. Decisions must be
made regarding the type of model computational methods, model extent, and amount of
1.3
1.4
1.5
1.6
CHAPTER 2
DESIGN CONSIDERATIONS AND REGULATORY REQUIREMENTS
2.1 INTRODUCTION
Hydraulic engineers and designers are faced with a wide variety of choices when
determining the capacity or location of a new bridge or an existing bridge that is to be
replaced. In addition to the choices regarding hydrologic and hydraulic components of a
bridge hydraulic analysis there are many other factors and requirements to consider.
One early consideration is the level of service the bridge is expected to provide. If the bridge
is remote and carries a low volume of traffic, it can be designed with a lower hydraulic
capacity resulting in a smaller and less expensive bridge. This means that the bridge and/or
approach roadways will be overtopped more frequently and the bridge owner can expect the
bridge and approach roadways to require more frequent maintenance and repair. On the
other hand, if the bridge is on an important route such that significant hardships or economic
impacts would be encountered if the bridge were out of service, then it should be designed
with a higher hydraulic capacity resulting in a larger and more expensive bridge and higher
approach embankments. These bridges and/or approach roadways would be rarely
overtopped and would need less frequent maintenance or repair. A smaller bridge may be
less expensive from a capital (initial) cost perspective, but this does not necessarily always
hold true from a life-cycle cost perspective. Most states or local jurisdictions have policies
and criteria that govern the level of service expected from their roadways and bridges.
There are also a significant number of permits that may be required when designing or
replacing a bridge. Federal, state, and local agencies have diverse and important interests
regarding the design and construction of bridges. A good hydraulic analysis conducted early
in the design process and a thorough understanding of the permitting and approval process
helps avoid costly redesigns or delays, and problems with permitting.
2.2 BRIDGE OPENING AND ROAD GRADE DESIGN CONSIDERATIONS
In general, given a particular design discharge at a given crossing, the shorter a bridge the
more backwater it will create. This same smaller bridge will also have higher velocities
through the bridge opening and an increased potential for scour at the bridge foundation. A
longer bridge at this same crossing will generate a smaller amount of backwater and will
have lower velocities and potential for scour. Policy considerations and economics require
an understanding of the impacts that the bridge could have on the flow of water in the
floodplain and impacts it might have on adjacent properties.
The bridge waterway width is directly associated with the bridge length, from abutment to
abutment. Hydraulic capacity should be a primary consideration in setting the bridge length.
The bridge must provide enough capacity to:
Freeboard refers to the vertical distance from the water surface upstream of the bridge to the
low chord of the bridge. The freeboard requirement is associated with a particular design
recurrence-interval event, which is usually the 50- or 100-year event. Rural, low-traffic routes
often allow a lower recurrence interval for establishing hydraulic capacity and freeboard.
2.1
The road profile can have a significant effect on bridge crossing hydraulics. Even if a bridge
is designed to provide freeboard above a 100-year flood, the approach roadways may be
overtopped by that same flood. When the overtopping occurs over a long segment of
roadway, the associated weir flow is an important component of the overall hydraulic
capacity of the crossing. In such a case, raising the road profile will have the potential to
increase backwater unless additional capacity is provided in the bridge waterway to
compensate for the lost roadway overtopping flow capacity.
The design of the piers and abutments has an effect on the bridge hydraulic capacity.
Although this effect is small compared to the bridge length and road profile, it can still be
important. For example, a bridge that crosses a regulatory floodway must be shown to cause
no increase in backwater over existing conditions. In such a case the energy losses that are
affected by the number of piers and their geometry can be significant. Spill-through
abutments, set well back from the tops of the main channel banks, are advisable when
bridge hydraulic capacity must be optimized.
Frequently the bridge waterway design includes subtle changes to the channel cross section
under the bridge and for a short distance upstream and downstream of the bridge. These
changes are intended to enhance channel stability and, in some cases, to improve hydraulic
efficiency. Channel stability can be enhanced, for instance, by grading the channel banks to
side slopes of 2H:1V or flatter, and by providing channel bank revetment. Capacity can be
improved by a moderate widening of the channel bottom in the immediate vicinity of the
bridge, with appropriate width transitions upstream and downstream.
There are several potential bridge opening and road grade considerations that impact
hydraulic capacity and upstream flood risk, especially when a road is upgraded and the
bridge is replaced. These include bridge length, deck width, abutment configuration (spill
through or vertical wall), number and size of piers, low chord elevation, freeboard, and road
grade. If a crossing with a 25-year level of service is improved to a 50-year level of service,
the road elevation may need to be increased. To avoid increased flood risk, the replacement
bridge may need to be considerably longer and higher than the existing bridge. If there is
inadequate freeboard, debris may collect along the deck and reduce flow conveyance.
2.3 FLOODPLAIN AND FLOODWAY REGULATIONS
A number of federal regulations affect the design and construction of bridges across the
nation's waterways. Executive Order (EO) 11988, which became law in 1977, is the source
from which the federal floodplain regulations are derived. The Federal Emergency
Management Agency (FEMA 2010) provides the following information regarding EO 11988.
Executive Order 11988 requires federal agencies to avoid to the extent
possible the long- and short-term adverse impacts associated with the
occupancy and modification of flood plains and to avoid direct and indirect
support of floodplain development wherever there is a practicable alternative.
In accomplishing this objective, "each agency shall provide leadership and
shall take action to reduce the risk of flood loss, to minimize the impact of
floods on human safety, health, and welfare, and to restore and preserve the
natural and beneficial values served by flood plains in carrying out its
responsibilities" for the following actions:
Acquiring, managing, and disposing of federal lands and facilities
2.2
improvements
Conducting federal activities and programs affecting land use, including but
not limited to water and related land resources planning, regulation, and
licensing activities
The guidelines address an eight-step process that agencies should carry out
as part of their decision-making on projects that have potential impacts to or
within the floodplain. The eight steps, which are summarized below, reflect the
decision-making process required in Section 2(a) of the Order.
1. Determine if a proposed action is in the base floodplain (that area which
has a one percent or greater chance of flooding in any given year).
2. Conduct early public review, including public notice.
3. Identify and evaluate practicable alternatives to locating in the base
floodplain, including alterative sites outside of the floodplain.
4. Identify impacts of the proposed action.
5. If impacts cannot be avoided, develop measures to minimize the impacts
and restore and preserve the floodplain, as appropriate.
6. Reevaluate alternatives.
7. Present the findings and a public explanation.
8. Implement the action.
Among a number of things, the Interagency Task Force on Floodplain
Management clarified the EO with respect to development in floodplains,
emphasizing the requirement for agencies to select alternative sites for
projects outside the floodplains, if practicable, and to develop measures to
mitigate unavoidable impacts.
FHWA regulations regarding the implementation of EO 11988 can be found in Title 23,
Section 650, Subpart A - Location and Hydraulic Design of Encroachments on Flood Plains
of the Code of Federal Regulations (23 CFR 650A). An FHWA policy statement referred to
as Non-Regulatory Supplement Attachment 2 provides additional guidance on complying
with the floodplain provisions of 23CFR650A. FEMA procedures for implementing this EO
are found in Title 44 Part 9 of the Code of Federal Regulations (44 CFR 9).
Floodplain regulations create constraints on the allowable backwater for the design of a new
bridge. The stringency of the constraint depends upon the status of the particular floodplain
being crossed. When a bridge project is to cross a FEMA floodplain featuring an established
regulatory floodway, the hydraulic engineer for the project must demonstrate that the fill
and/or bridge elements to be constructed within the floodway will not cause any increase in
the 100-year flood water surface elevation compared to existing conditions. This constraint is
often termed a no-rise requirement. If meeting the no-rise requirement is not practicable, the
bridge owner must coordinate with the local community floodplain administrator and with
FEMA to revise the floodplain mapping and floodway boundaries as appropriate. In such a
case the local community could be sanctioned or penalized by FEMA under the National
Flood Insurance Program unless a Conditional Letter of Map Revision (CLOMR) request is
submitted to and approved by FEMA prior to the beginning of project construction.
2.3
When crossing a FEMA floodplain without a regulatory floodway, federal regulations are less
stringent. In such a case the federal regulations allow the project, combined with existing
development that has occurred since the floodplain map became effective, and with other
future developments that might reasonably be anticipated, to cause up a 1.0 foot increase in
the 100-year flood water surface elevation. In many locations, however, state regulations or
local ordinances may impose a more stringent constraint than the federal regulations.
Some bridge projects involve replacing an existing floodplain crossing that causes significant
backwater over pre-bridge conditions. If the bridge to be replaced is known to cause more
than 1 foot of backwater, it is advisable to design the replacement bridge to avoid any
additional backwater, even if the floodplain regulations might allow a moderate increase. In
such a case the new bridge should result in some reduction of the backwater.
Even though floodplain regulations are derived from federal laws, floodplain management is
a function of state or local government. The project-specific enforcement of floodplain
regulations, therefore, typically takes the form of floodplain permits from state or local
agencies. It is the responsibility of the bridge owner to assure that any potential designs meet
the criteria outlined in these regulations and assure that all required floodplain permits are
applied for and received before the construction of a new or replacement bridge takes place.
2.4 SCOUR AND STREAM STABILITY CONSIDERATIONS AND GUIDANCE
Another critical component of the design and/or evaluation of a bridge opening is to design
the bridge to be stable from scour at the piers, abutments, and across the contracted
opening. From a hydraulic perspective, the magnitude of local scour at a pier is a function of
depth and velocity of flow, alignment of the pier with flow, and pier type and location.
Depending of foundation costs and complexity it will be necessary to balance the number
and size of piers, length and height, and anticipated total scour depth against increased
costs of the superstructure associated with longer spans (girder type and allowable span)
and foundation required to resist scour.
The magnitude of local scour at an abutment is a function of depth and velocity of flow, the
skew of the embankment to the floodplain, as well as the amount of flow from the overbank
that passes through the bridge opening. It is also a function of where the abutment is located
in relation to the main channel. It is recommended that an abutment not be located in or
close to the main channel if at all possible.
The amount of contraction scour that occurs at a bridge crossing is a function of the degree
that a bridge contracts floodplain flow. In general, bridges with higher degrees of contraction
can be expected to have higher flow velocities and larger scour depths. If the depths of
contraction scour are too large it may be necessary to increase the bridge length to reduce
scour across the bridge opening.
Bridges should be designed to withstand scour from large floods and from stream instabilities
expected over the life of a bridge. Recommended procedures for evaluating and designing
bridges to resist scour can be found in FHWA publications HEC-20 (FHWA 2012a) and HEC18 (FHWA 2012b).
2.4
2.5
2.6
CHAPTER 3
GOVERNING EQUATIONS AND FLOW CLASSIFICATION
3.1 INTRODUCTION
This chapter provides background on the fundamentals of rigid boundary open channel flow.
Although this is not a hydraulic engineering textbook, there is sufficient information to act as
a source reference on the equations used in open channel and bridge hydraulics. In open
channel flow the upper surface of the water is in contact with the atmosphere, therefore, the
surface configuration, flow pattern and pressure distribution depend primarily on gravity.
Because the flow involves a free surface, it has more degrees of freedom than flow in a
closed conduit flowing full. The types of flow include:
The following sections will emphasize open channel flow as being: (1) uniform or nonuniform
(varied) flow; (2) steady or unsteady flow; (3) laminar or turbulent flow; and (4) subcritical or
supercritical.
3.1.1 Streamlines and Streamtubes
The motion of each fluid particle is described in terms of its velocity vector, V, which is
defined as the time rate of change of the position of the particle. The particle's velocity is a
vector quantity with a magnitude (the speed, V = |V|) and direction. As the particle moves, it
follows a particular path, which is governed by the velocity of the particle. The location of the
particle along the path is a function of where the particle started at the initial time and its
velocity along the path. If the flow is steady (i.e., nothing changes with time at a given
location in the flow field), each successive particle that passes through a given point such as
point (1) in Figure 3.1, will follow the same path. For such cases the path is a fixed line in the
X-Z plane. Neighboring particles that pass either side of point (1) follow their own paths,
which may be of different shape but do not cross the one passing through (1). The entire XZ plane is filled with such paths.
For steady flow each particle progresses along its path and its velocity vector is everywhere
tangent to the path. The lines that are tangent to the velocity vectors throughout the flow
field are called streamlines. For many situations it is easiest to describe the flow in terms of
the "streamline" coordinates based on the streamlines as shown in Figure 3.1. The particle
motion is described in terms of its distance along the streamline. The distance along the
streamline is related to the particle speed by V = ds/dt, and the radius of curvature is related
to the shape of the streamline. In addition to the equal potential coordinates along the
streamlines, the coordinate normal to the streamline, n, will be of use in the applications of
open channel flow.
3.1
Streamlines
V
V
Figure 3.1. Flow in the X-Z plane and flow in terms of streamline and normal coordinates.
3.1.2 Definitions
Velocity: The velocity of a fluid particle is the rate of displacement of the particle from one
point to another and is a vector quantity having both magnitude and direction. The
mathematical formulation of velocity magnitude is given in Equation 3.1.
ds
dt
V=
(3.1)
Streamline: A streamline is an imaginary line within the flow that is tangent everywhere to
the velocity vector, see Figure 3.1. Since the flow is tangent to the streamline, there cannot
be any net movement of fluid across the streamline in any direction.
Streamtube: A streamtube is an element of fluid bounded by a pair of streamlines that
enclose or confine the flow. Since there can be no net movement of fluid across a
streamline, it follows that there can be no net movement of fluid in or out of the streamtube,
except at the ends. This fact will be utilized in the development of the continuity equation.
Acceleration: Acceleration is the time rate of change in magnitude or direction of the
velocity vector. Acceleration can be expressed by the total derivative of the velocity vector
as follows:
a=
dv
dt
(3.2)
The vector acceleration, a, has components both tangential and normal to the streamline, the
tangential component representing the change in magnitude of the velocity, and the normal
component reflecting the change in direction:
as =
dv s v s
( v s2 )
=
+ 1/ 2
dt
t
s
(3.3)
3.2
an =
dv n v n ( v 2 )
=
+
dt
dt
r
(3.4)
The first terms in Equations 3.3 and 3.4 represent the change in velocity, both magnitude
and direction, with time at a given point. This is called local acceleration. The second term
in each equation is the change in velocity, both magnitude and direction, with distance. This
is called convective acceleration.
3.1.3 Classification of Open Channel Flow
Uniform flow in open channels depends upon there being no change with distance in either
the magnitude or the direction of the velocity along a streamline; that is both v/s = 0, and
v/n = 0. Nonuniform flow in open channels occurs when either v/s 0 or v/n 0. The
particular type of nonuniform flow that occurs when v/s 0 in open channels is usually
called varied flow. Figure 3.2 illustrates uniform flow in a straight channel having a constant
depth of flow, a constant slope, and a constant cross section throughout. Obviously, this
condition seldom exists in nature. Examples of nonuniform flow, where v/n 0, are bends
or curving sides of the channel. When v/s 0, the flow is varied and occurs when there is
a change in depth of flow due either to a change in slope, a barrier or drop, or a change in
side or bottom, so that the velocity increases or decreases in the direction of flow, Figure 3.3.
Y2
Y1
3.3
Y2
Y1
Figure 3.3. Example of nonuniform flow where the depth of flow Y2 Y1.
Unlike laminar and turbulent flow, subcritical and supercritical flows exist only with a free
surface or interface. The criterion for subcritical (tranquil) and supercritical (rapid) flow is the
V
VD
, which like the Reynolds Number, R e =
, is the ratio of two
Froude Number, FR =
gy
types of forces.
The Froude number is a ratio of the forces of inertia to the forces of gravity and is discussed
in detail later in this chapter. It will suffice at this point to indicate that when the FR = 1 the
flow is critical, when FR 1 the flow is tranquil, and when the FR 1 the flow is rapid.
In the above discussion, there are four classifications needed to describe the type of flow in
an open channel.
1.
2.
3.
4.
Uniform or nonuniform
Steady or unsteady
Laminar or turbulent
Subcritical or supercritical
One from each of these four types must exist. Because the classifying characteristics are
independent, sixteen different types of flow can occur. These terms, uniform or nonuniform,
steady or unsteady, laminar or turbulent, subcritical or supercritical, and the two
dimensionless numbers (the Froude number and the Reynolds number) are more fully
explained in the following sections.
3.2 THREE BASIC EQUATIONS OF OPEN CHANNEL FLOW
The basic equations of flow in open channels are derived from the three conservation laws.
These are: (1) the conservation of mass, (2) the conservation of energy, and (3) the
conservation of linear momentum. The conservation of mass is another way of stating that
(except for mass-energy interchange) matter can neither be created nor destroyed. The
conservation of energy is an empirical law of physics that in a closed system the energy
remains constant over time. Similar to the conservation of mass, energy can neither be
3.4
created nor destroyed, although it can be transformed from one state to another (i.e., kinetic
energy to potential energy). The principle of conservation of linear momentum is based on
Newton's second law of motion which states that a mass (of fluid) accelerates in the direction
of and in proportion to the applied forces on the mass.
In the analysis of flow problems, much simplification can result if there is no acceleration of
flow or if the acceleration is primarily in one direction and the accelerations in other directions
are negligible. However, a very inaccurate analysis may occur if it is assumed that
accelerations are small or zero when in fact they are not. The concepts in this chapter
assume one-dimensional flow and the derivations of the equations utilize a control volume
concept. A control volume (Figure 3.4) is a volume which is fixed in space or moving with the
fluid and through whose boundary matter, mass, momentum, energy can flow. The volume
is called a control volume and its boundary is a control surface.
Section 1
Section 2
=0
Mass can enter or leave the control volume through any or all of the control volume surfaces.
Rainfall would contribute mass through the surface of the control volume and seepage
passes through the interface between the water and the banks and bed. In the absence of
any lateral mass fluxes, the mass enters the control volume at section 1 and leaves at
section 2, or
Dm
=0
Dt
(3.5)
3.5
The fluid mass can be represented by the density times the volume, dxdydz, and Equation
3.5 can be written as:
d
dt
dxdydz +
CVol
dxdydz = 0
(3.6)
CS
Orienting the axes for sections one and two to be perpendicular to dx, then
dx/dt = Velocity in the x direction
(3.7)
In the absence of any lateral mass fluxes, the mass enters the control volume at section 1
and leaves at section 2. Further assuming steady flow, Equation 3.6 can be written as:
A1
dx
dydz =
dt
1
dx
dydz
2
dt
A2
(3.8)
Substituting Equation 3.7 into 3.8 the continuity equation reduces to inflow equal outflow.
A1
1 ( v 1 n1 )dA 1 =
A2
2 ( v 2 n 2 )dA 2
(3.9)
Where n is the outward normal unit vector, and A is the area of the control surface that the
flow is passing through. The dot product of the velocity vector with the unit outward normal
(V x n) determines the component of the velocity perpendicular to the surface since only that
component can carry mass through the surface. For most open channel flow situations flow
is virtually incompressible and the equation reduces to:
A1
( v 1 n1 )dA 1 =
A2
( v 2 n 2 )dA 2
(3.10)
It is often convenient to work with the average conditions at a cross section, so an average
velocity V is defined such that
V=
1
A
vdA
(3.11)
The symbol v represents the local velocity whereas V is the average velocity at the cross
section. Therefore, for steady incompressible flow the continuity equation can be reduced to
A 1V1 = A 2 V2 = Q = AV
(3.12)
where Q is the volume flow rate or the discharge. Equation 3.12 is the familiar form of the
conservation of mass equation for steady flow in open channels. It is applicable when the
fluid density is constant, the flow is steady and there is no significant lateral inflow or
seepage. The velocity will generally vary in both direction and magnitude over the cross
section and the summation (integral) of the area over the cross section must be at right
angles to the velocity component.
3.6
Two-Dimensional Form of the Continuity Equation. Considering flow through a small control
volume as shown in Figure 3.5, assume a general flow, V(x,y,z,t). Flow through surfaces 1
and 2 are perpendicular to the Y-Z plane. Note that the efflux rate through area 1 is -Vx per
unit area and the flow through the area is given as Vx + {(Vx)/x}dx. Therefore, the net
efflux through the surface would be {(Vx)/x}dx. Performing similar computations for the
other sides and adding the results the total net efflux rate is:
=
y
z
t
x
And for steady incompressible flow = constant and /t = 0, which gives the simplified
differential form of the continuity equation.
( Vx ) ( Vy ) ( Vz )
+
+
=0
y
z
x
The equation states that for steady flow the rate of flow into the control volume must be equal
to the rate of flow out.
3.7
P
Py dx dz +
dy dx dz
y
P
Px dy dz +
dx dy dz
x
Fy
Px dy dz
X
Py dx dz
Figure 3.6. Surfaces forces acting on a fluid element in the X and Y directions for an
inviscid fluid.
3.8
The forces tending to accelerate the particle are: pressure forces on the ends of the system,
pdA - (p+dp)dA = dpdA (the pressure on the sides of the system have no effect on its
acceleration), and the component of weight in the direction of motion, -gn dsdA (dz / ds = gndAdz. The differential mass being accelerated by the action of these differential forces is
dM = dsdA. Applying Newton's second law, dF = (dM)a, along a streamline and using the
dv
gives:
one dimensional expression for acceleration for steady flow, a s = v
ds
dpdA gn dAdz = (dsdA)v
dv
ds
(3.13)
(3.14)
2gn
(3.15)
p v2
d +
+ z = 0
2gn
(3.16)
For incompressible flow of a uniform density fluid, the one-dimensional Euler equation can be
integrated between any two points (because and gn are both constant) along a streamline
to obtain the Bernoulli equation
p1
V2
p
V2
+ 1 + z1 = 2 + 2 + z 2
2gn
2gn
(3.17)
Equation 3.17 applies to all points on the streamline and thus provides a useful relationship
between pressure, the magnitude of the velocity, and the height above the datum. An
empirical relationship can be added to account for the losses in the system. These losses
include friction and transition (expansion and contraction) losses that are described in
Chapter 5.
Velocity Distribution. As shown in Figure 3.7 the requirement of zero velocity at a boundary
for either laminar and turbulent flow produces velocity distributions that are not uniform
(nonuniform). The term v2/2g is the kinetic energy per unit weight at a particular point. If the
velocity distribution varies across the section of the flow, the total kinetic energy of the
section will be greater than the kinetic energy computed from the average velocity (e.g., the
average value of the sum of incremental velocity squared is greater than the average velocity
squared).
3.9
dQ
v
A
dA
1
A
Q
A
v dA =
The total kinetic energy per unit time that passes the section is determined by integrating the
product of the kinetic energy per unit weight and the weight of the fluid passing per unit time
from streamline to streamline across the section.
Energy flux =
v2
( dQ) =
2g
v2
v dA =
2g
2g
v 3 dA
(3.18)
The energy flux using the average velocity would then require a correction coefficient .
Energy flux =
A V3
2g
(3.19)
Solving for the energy correction coefficient yields the following relationship:
1
A
v3
3 dA
V
(3.20)
3.10
Flux of
momentum
into the control
volume
Time rate of
change of
momentum in the
control volume
The statement of
The terms in the statement are vectors so direction as well as magnitude must be
considered. Consider the conservation of momentum in the direction of flow (the x-direction
in Figure 3.8). At the outflow section (section 2), the flux of momentum out of the control
volume through the differential area dA2 is:
2 v 2 dA 2 v 2
(3.21)
1
dx
v1
P1
A1
WP1
A2
v2
WP2
P2
A2
2 v 2 dA 2 v 2
(3.22)
Similarly, at the inflow section (section 1), the flux of momentum into the control volume is:
A1
1v 1dA 1v 1
(3.23)
3.11
Vol
{
t
vd ( Vol)}
(3.24)
Vol
At the upstream section, the force acting on the differential area dA1 of the control volume is
p1dA1, where p1 is the pressure from the upstream fluid on the differential area. The total
force in the x-direction at section 1 is
p 2 dA 2 . There is also a fluid shear stress o acting along the interface between the water
2
and the bed and banks (over the wetted perimeter). The shear on the control volume is in
the direction opposite to the direction of flow and results in a force -oWPdx where o is the
average shear stress on the interface area, and WP is the average wetted perimeter and dx
is the length of the control volume. The WPdx is the interface area (the area that the water is
in contact with).
The forces affecting the body fall into two classes, surface forces (as were just identified) and
body forces. Another surface force not included in this derivation is a wind stress acting on
the water surface. Body forces are forces with a long range of influence which act on all the
material particles in the body and which, as a rule, have their source in fields of force. The
most important example of a body force is the earth's gravity field. The body force
component in the x-direction is denoted by Fb and will be discussed in a subsequent section.
The statement of conservation of momentum in the x-direction for the control volume is:
A2
2 v 22 dA 2
A1
1v 12 dA 1 +
vd( Vol) =
Vol
A1
p1dA 1
A2
p 2 dA 2
o WPdx + Fb
(3.25)
As with the conservation of mass equation, it is convenient to use average velocities instead
of point velocities. The momentum coefficient is defined so that when average velocities
are used instead of point velocities, the correct momentum flux is considered.
=
1
V2A
v 2 dA
(3.26)
For steady incompressible flow, Equation 3.25 is combined with Equation 3.26 to give:
2 V22 A 2 1V12 A 1 =
A1
p1dA 1
A2
p 2 dA 2
o WPdx + Fb
(3.27)
The pressure force and shear force terms on the right-hand side of Equation 3.27 are usually
abbreviated as Fx so:
F =
x
A1
p1dA 1
A2
p 2 dA 2
o WPdx + Fb
3.12
(3.28)
2 V22 A 2 1V12 A 1 =
(3.29)
For steady flow with constant density, combining Equation 3.12 with 3.29, the steady flow
conservation of linear momentum equation takes the familiar form:
Q( 2 V2 1V1 ) =
(3.30)
Depending on the situation, other external forces need to be applied, such as a surface force
due to wind blowing over the control volume. This force plays a significant role when
analyzing currents during a hurricane storm surge.
3.3 FLOW RESISTANCE AND OTHER HYDRAULIC EQUATIONS
3.3.1 Flow Resistance
This section provides basic information to determine the friction loss coefficients for steady
flow in natural channels. The two most commonly used equations for the computation of
steady flow in natural channels are the Chezy and Manning equations.
Development of the Manning Formula. In 1889 Manning developed an equation to estimate
flow in an open channel as an alternative to measuring flow that passes over or through a
hydraulic structure such as a weir or flume. He assumed that the flow was uniform. For this
condition, the forces acting on a control volume (Figure 3.9) can be quantified and used to
develop flow resistance equations.
Fx = m a
2
FP2
FW(x)
Ff
Z2
FP1
L
Z1
Datum
Figure 3.9. Forces acting on a control volume for uniform flow conditions.
3.13
Summing the forces acting on the control volume in the x direction gives the following:
= ma
(3.31)
Noting that the mass times acceleration term is zero for uniform flow conditions (i.e., the
velocity at section 1 and is the same as the velocity at section 2, therefore there is no
acceleration of the fluid) Equation 3.31 becomes:
Fp1 Fp 2 + Fw ( x ) Ff = 0
(3.32)
where:
Fp1 and Fp2
Fw(x)
Ff
L
Z
=
=
=
=
=
=
Since the depth of flow is the same at sections 1 and 2 and the cross sections are the same,
the hydrostatic forces at sections 1 and 2 are equal and opposite in sign.
Fx1 = Fx 2
(3.33)
Fw Ff = 0
(3.34)
( A 1 + A 2 )
LSo , weight of water multiplied by bottom of the bed slope, and
2
Ff = (WP)L. Substituting these expressions for Ff and Fw, yields:
Noting that Fw ( x ) =
( WP)L =
( A 1 + A 2 )
L
2
(3.35)
A
So
WP
(3.36)
where:
A is the average area between sections 1 and 2, also note that A1 = A2 uniform flow. Flow
velocity in the channel depends on its cross-sectional shape (among other factors), and the
resistance to the flow depends upon the shear stress acting over the channel boundary, the
wetted perimeter. The hydraulic radius is defined as the ratio of the channel cross-sectional
area to the channel wetted perimeter (the portion in contact with the flow R = A/WP and is
defined as the hydraulic radius. Rewriting Equation 3.36 yields.
3.14
= RS o
(3.37)
For the remainder of this section the subscript on slope will be dropped and only use the
symbol S, remembering for uniform flow, Sf = Sw = So. Therefore, Equation 3.37 can be
expressed as:
= RS
(3.38)
fV 2
.
8
Darcy, Weisbach and others developed an expression for the head loss in a long straight
cylindrical pipe that is a function of the friction factor, f, of the pipe boundary, the diameter of
the pipe, D, the length of the between points of interest (i.e., distance between 1 and 2), and
2
the velocity head, V /2g and proposed an equation of the form:
From fundamental fluid mechanics for pipe flow it is assumed that the shear stress =
hf = f
L V2
= SL
D 2g
(3.39)
where:
hf
f
L
D
=
=
=
=
2g
DS
f
(3.40)
Noting that D = 4 R for a cylindrical pipe flowing full and substituting 4R for the diameter of
the pipe gives:
V=
8g
f
(3.41)
RS
In 1775 Chezy established his relationship that identified the Chezy Coefficient C to equal
8g
and published his equation for uniform flow as:
f
V = C RS
(3.42)
3.15
C=
1 1/ 6
R
n
(3.43)
1 2 / 3 1/ 2
R S
n
(3.44)
Manning developed his formula in the metric system with the unit of length being the meter.
To convert Equation 3.44 to U.S. Customary units and maintain the value of n in both
systems, a factor of 1.486 needs to be included in the equation. Therefore, the Manning
equation in U.S. Customary units becomes:
V=
1.486 2 / 3 1/ 2
R S
n
(3.45)
The 1.486 is due to the dimensional relationship of the equation (1 / n = t 1L1/ 3 ) and converting
meters to feet (3.28081/3 = 1.486).
By applying the continuity equation, the Manning equation can be written in terms of
discharge as:
Q=
1.486
AR 2 / 3 S1/ 2
n
(3.46)
n = (no + n1 + n 2 + n3 + n 4 )m5
(3.47)
Where no is a base n value for a straight, uniform, smooth channel, n1 is the degree of
surface irregularities of the channel, n2 is the variation of the channel cross section, n3 is the
relative effect of obstructions, n4 is due to the effect of vegetation and flow conditions, and m5
relates to the degree of meandering. Table 3.1 is a reproduction of Cowan's summary table
taken from Chow (1959). Chow also presented an excellent table listing typical n values for
a range of conditions. The minimum, normal, and maximum values of n are shown in the
table. A more complete discussion can be found in Chow (1959, pp. 108-113). Table 3.2
shows a portion of the table for Natural Streams taken from Chow's Open-Channel
Hydraulics book.
3.16
Low
n4
0.005-0.010
Vegetation
Medium
n4
0.010-0.025
Vegetation
High
n4
0.025-0.050
Vegetation
Very High
n4
0.050-0.100
Degree of meandering
Minor
m5
1.000
Degree of meandering
Appreciable
m5
1.150
Degree of meandering
Severe
m5
1.300
3.17
Table 3.2. Values of the Manning Roughness Coefficient for Natural Channels.
Type of channel and description
Minimum
Normal
Maximum
D. Natural Streams
D-1 Minor stream (top width at flood stage < 100 ft)
blank
blank
blank
blank
blank
blank
a. Stream on plain
1. Clean, straight, full stage, no rifts or deep pools
2. Same as above, but more stones and weeds
3. Clean, winding, some pools and shoals
4. Same as above, but some weeds and stones
5. Same as above, lower stages, more ineffective slopes and
sections
6. Same as above, but more stones
7. Sluggish reaches, weedy, deep pools
8. Very weedy reaches, deep pools, or floodways with heavy
stand of timber and underbrush
b. Mountain streams, no vegetation in channel, banks usually
steep, trees and brush along banks submerged at high stages
1. Bottom: gravels, cobbles, and few boulders
2. Bottom: cobbles with large boulders
blank
0.025
0.030
0.033
0.035
blank
0.030
0.035
0.040
0.045
blank
0.033
0.040
0.045
0.050
0.040
0.048
0.055
0.045
0.050
0.050
0.070
0.060
0.080
0.075
0.100
0.150
blank
blank
blank
0.03
0.04
0.04
0.05
0.05
0.07
blank
blank
0.025
0.030
blank
blank
0.030
0.035
blank
blank
0.035
0.050
b. Cultivated areas
1. No crop
2. Mature row crops
3. Mature field crops
blank
0.020
0.025
0.030
blank
0.030
0.035
0.040
blank
0.040
0.045
0.050
c. Brush
1. Scattered brush, heavy weeds
2. Light brush and trees, in winter
3. Light brush and trees, in summer
4. Medium to dense brush, in winter
5. Medium to dense brush, in summer
blank
0.035
0.035
0.040
0.045
0.070
blank
0.05
0.05
0.06
0.07
0.10
blank
0.07
0.06
0.08
0.11
0.16
d. Trees
1. Dense willows, summer, straight
2. Cleared land with tree stumps, no sprouts
3. Same as above, but with heavy growth of sprouts
4. Heavy stand of timber, a few down trees, little
undergrowth, flood stage below branches
5. Same as above, but with flood stage reaching branches
blank
0.11
0.03
0.05
blank
0.15
0.04
0.06
blank
0.20
0.05
0.08
0.08
0.10
0.12
0.10
0.12
0.16
D-3 Major streams (top width at flood stage > 100 ft). The n value
is less than that for minor streams of similar description,
because banks offer less effective resistance.
blank
blank
blank
a.
0.025
up to
0.06
b.
0.035
up to
0.10
3.18
Other sources for determining the Manning Roughness factors include pictures of selected
streams and over-bank floodplains to use as a guide for selecting n. The publication "Guide
for Selecting Manning's Roughness Coefficients for Natural Channels and Flood Plains"
(FHWA 1984b) is an excellent resource on how to estimate Manning n values for both
channel and out-of-bank flows. The U.S. Geological Water Supply Paper 1849, "Roughness
Characteristics of Natural Channels," (USGS 1967) is another example of color photographs
of natural channels exhibiting various values of n. These publications can be found on the
internet by searching the document titles. Other publications that have photographs of
calibrated streams are: Arcement and Schneider (USGS 1989), Chow (1959), Hicks and
Mason (1991) and NRCS (1963).
Figure 3.10 shows a floodplain with a computed roughness coefficient: Manning n of 0.20.
The vegetation of the floodplain is a mixture of small and large trees, including oak, gum, and
ironwood. The base is firm soil and has minor surface irregularities. Obstructions are minor.
Ground cover is medium, and there is a large amount of undergrowth that includes vines and
palmettos (FHWA 1984b). Similarly, Figure 3.11 is taken from the USGS (1967) Water
Supply Paper and shows a channel with a computed Manning n of 0.026.
3.19
Figure 3.11. Looking upstream from the right bank on Indian Fork, near
New Cumberland, Ohio (USGS 1967).
3.3.2 Drag Force
As stated earlier, resistance to flow can be divided into shear resistance and resistance due
the difference in pressure from the upstream side to the downstream side of an object (form
drag). When flow passes over an object a resistance to flow is created that depends upon
the shape or form of the boundary of the object. The shape of the boundary (i.e., a bridge
pier) causes a deflection of the streamlines and local acceleration of the fluid. Consequently
a change in pressure takes place from the upstream to the downstream side of the boundary,
which is also referred to as a normal stress. The summation of the forces over the surface
results is a drag force on the boundary and a pressure resistance against the fluid.
Derivation of the Drag Equation. The determination of the drag of a flowing fluid past a
boundary can be accomplished using dimensional analysis to determine the significant
variables and through experimental data to determine the numerical relationships between
the parameters. For incompressible flow the drag force can be written as a function of the
following parameters that represent the geometry of the object (area, A), the flow (velocity of
flow, V), the roughness of the boundary (roughness height, e) and the fluid (density and
viscosity of the fluid, and ):
FD = func ( A, V, , , e)
(3.48)
3.20
The Buckingham method of dimensional analysis shows that three non-dimensional groups
can be formed. Choosing A, V, and as the repeating independent variables yields the
following relationship.
A 1/ 2V e
FD = func AV 2 ,
,
(3.49)
Equation 3.48 can be rearranged where FD/AV2 is defined as the coefficient of drag CD.
FD
e
= CD = func R e ,
2
AV
L
(3.50)
Since stagnation pressure can be represented as V2/2, Equation 3.49 can be rearranged as
follows:
FD / A
e
= CD = func R e ,
2
V
L
(3.51)
AV 2 1
= CDAV 2
2
2
(3.52)
The force of drag then is a function of the area of the object, the velocity of the flow, and the
density of the fluid. The drag coefficient CD depends upon the Reynolds Number Re which in
turn depends upon the inertial effects of the flow relative to the viscous effects. The
coefficient of drag for many objects has been determined experimentally.
Application of Drag to Piers. The placement of bridge piers in the channel or the floodplain of
natural rivers will cause an additional backwater due to the pier obstruction to the flow. The
drag force created by the pier can be computed using Equation 3.52 and knowing the
coefficient of drag CD. Lindsey (1938) provided drag coefficients for various objects as a
function of Reynolds Number. The drag coefficient is dependent on the ratio of pier area to
the total area of the bridge opening, the type of piers, pier shape, and the orientation of flow.
Table 3.3 includes typical drag coefficient of piers as given in the HEC-RAS Reference
Manual (USACE 2010c).
Table 3.3. Typical Drag Coefficients for Different Pier Shapes.
Pier Shape
Circular pier
Elongated piers with semi-circular ends
Elliptical piers with 2:1 length to width
Elliptical piers with 4:1 length to width
Elliptical piers with 8:1 length to width
Square nose piers
Triangular nose with 30 degree angle
Triangular nose with 60 degree angle
Triangular nose with 90 degree angle
Triangular nose with 120 degree angle
3.21
Drag Coefficient CD
1.20
1.33
0.60
0.32
0.29
2.00
1.00
1.39
1.60
1.72
Drag Force due to the Addition of Debris. The accumulation of debris on bridge piers as
illustrated in Figure 3.12 and on the superstructure can create significant forces on the
structure. The hydraulic effects of debris can be analyzed using a one-dimensional model.
However, depending upon the complexity of the hydraulics and the risk of failure of the
structure, two- or three-dimensional models as well as physical model studies can be
performed. The reader is referred to NCHRP Report 445, Debris Forces on Highway Bridges
(NCHRP 2000), and Hydraulic Engineering Circular No. 9 (FHWA 2005), Debris Control
Structures Evaluation and Countermeasures, Chapter 4 Analyzing and Modeling Debris
Impacts to Structures.
Figure 3.13. An example of upstream bridge cross section with debris accumulation
on a single pier.
3.22
Velocity
Distribution
Crest
H
nappe
3.23
Writing Bernoulli's equation between section 1, which is in the approach channel where the
velocity profile is uniform and section 2, which is slightly downstream of the weir crest along
the streamline AB, results in the following relationship:
H+
V12
V2
= (H h) + 2
2g
2g
(3.53)
2
B
EGL
HGL
v2
v1
P
A
Figure 3.15. Example of simplified weir flow for a sharp crested weir.
Solving for the velocity at section 2 gives:
V2
V2 = 2g h + 1
2g
(3.54)
To solve for the flow rate per unit width q = V2H leads to the following equation:
q=
V2 dh = 2g
V2
h + 1
2g
1/ 2
(3.55)
dh
V 2 V 2
2
2g H + 1 1
3
2g 2g
3/2
(3.56)
Typically Equation 3.56 can be simplified by assuming that V1 is small and therefore V12/2g
can be neglected. The kinetic energy term can be neglected for approach velocities of 2 ft/s
(0.6 m/s) or less and for most practical problems there is more uncertainty in determining the
appropriate coefficient. Therefore, Equation 3.56 can be written to solve for unit discharge,
q:
q=
2
2g H3 / 2
3
(3.57)
3.24
Equation 3.57 is the basic equation for the typical rectangular weir. Because of the
assumptions made in developing the simplified form of the weir equation a coefficient needs
to be introduced to account for the properties that were neglected and therefore, Equation
3.57 can be expressed as:
q = Cw
2
2g H3 / 2
3
(3.58)
Q = C w LH3 / 2
(3.59)
There has been a significant amount of research to determine the coefficient for the various
types of weirs that have been developed. Note that Cw in Equation 3.59 includes
gravitational acceleration so the value depends on the system of units being used.
Ogee Spillway Crest. Diversion structures are used to divert water from an existing natural
watercourse into an off-channel conveyance system. Very often the shape is that of an ogee
weir (Figure 3.16). The ogee spillway is designed for a specific flow, Q0, and a specific head,
H0, and therefore there is a unique coefficient C0 for this flow. Any other flow and head will
have a different coefficient. The United State Bureau of Reclamation in their Design of Small
Dams (USBR 1987) has design curves for these conditions (Figure 3.17).
When flow overtops a weir and there is significant tailwater the discharge will be reduced due
to the submergence of the weir. For an ogee shape weir, the USBR (1987) has developed a
relationship that is presented in Figure 3.18 to account for the reduction in the discharge
coefficient, which in turn reduces the flow over the weir.
H0
3.25
Figure 3.17. Discharge coefficients for vertical-face ogee crest (USBR 1987).
Figure 3.18. Ratio of discharge coefficients caused by tailwater effects (USBR 1987).
3.26
Broad-Crested Weir. The flow over a broad-crested weir will occur at critical depth for an
ideal fluid flow (Figure 3.19). The flow over a broad-crested weir can be computed using the
continuity equation and assuming that there is a hydrostatic pressure distribution where
critical depth yc occurs. For a rectangular channel the Froude Number at minimum energy
(critical depth) is equal to one.
FR = 1 =
Vc
gy c
V12 / 2g
yc
Q = L y c gy c
Also noting that for minimum energy occurring in a rectangular channel the critical depth is
equal to 2/3 of the head H. Substituting yields:
2
Q = L g H
3
3/2
= 3.089 L H3 / 2
(3.60)
The coefficient for SI units is 1.705. In many situations the critical depth section may be in a
area of strong streamline curvature and the boundary friction along the crest may reduce the
true specific energy from the assumed value of H by the time the flow reaches the critical
section. Therefore, the more general form of the equation for a broad-crested weir is:
Q = C L H3 / 2
(3.61)
The coefficient is a function of P/H, L/H, Re, shape of the weir, and roughness. Discharge
coefficients for a broad-crested weir usually range from about 2.6 to 3.05 (1.44 to 1.68 for
SI). Flow over a bridge and the roadway approaches (embankments) is usually calculated
using the general broad-crested weir equation.
3.27
Weir Coefficient. The discharge coefficient for a broad-crested weir as stated above ranges
between 2.6 and 3.05 (1.44 to 1.68 for SI). Flow overtopping a bridge deck is not an ideal
broad-crested weir and it is generally recommended that the lower value be used for the
discharge coefficient where increased resistance to flow caused by obstructions such as
bridge railings, curbs, and debris would cause a decrease in the value of C. From King's
Handbook (King and Brater 1963), weir coefficients are given with respect to the head on the
weir and to the width of weir. The Hydraulics of Bridge Waterways manual (FHWA 1978)
provides a curve of C versus head on the roadway. "Hydraulic Performance of Bridge Rails"
(Charbeneau et al. 2008) shows the impact of bridge rails on the computation for the
overtopping flow.
Similar to the Ogee Weir, when flow overtops a broad-crested weir and there is significant
tailwater the discharge will be reduced due to the submergence of the weir. For broadcrested weirs FHWA (1978) has developed a relationship to account for weir submergence
(Figure 3.20).
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
76
78
80
82
84
86
88
90
92
94
96
98
100
Percent Submergence
Figure 3.20. Discharge reduction factor versus percent of submergence.
3.3.4 Gate and Orifice Equations
This section deals with inline structures that are commonly found in open channel flow.
Common examples of inline structures are the orifice and Tainter gates (radial gates). Other
inline structures such as bridges can often be modeled with an orifice type equation.
Sluice and Tainter Gates. The two most important design features for the sluice (vertical lift)
and Tainter gates (Figure 3.21) are the head (elevation) versus the discharge relationship
and the pressure distribution over the gate surfaces. The structural design of the orifice
involves consideration of the hydrostatic force on the gate, the hoisting force, the weight of
the gate, and the roller friction (the friction on the gate is reduced by rollers that are typically
attached to the gate). The Tainter gate has a circular segment for its face which rotates
about the center of the curvature. Since the hydrostatic pressures are radial, passing
3.28
through the trunnion bearing, the thrust on the gate is substituted for the roller friction of the
orifice. The pin friction is usually much less than the roller friction, so that the Tainter gate is
comparatively light and easy to operate.
For the sluice and Tainter gates shown in Figure 3.21, the Bernoulli equation for onedimensional flow can be used to solve for the discharge.
V22
V2
+ y 2 = 1 + y1
2g
2g
(3.62)
V22 / 2g
E
V22 / 2g
E
V12 / 2g
Y2
V12 / 2g
Y2
h
Y1
Y1
Tainter Gate
Sluice Gate
(3.63)
Then solving for the discharge gives the relation for flow passing under the gate as:
V22
Q = C c hL 2g( y 2 y 1 ) +
2g
(3.64)
Although this derivation ignored the losses due to the boundary development along the gate
and floor, these are usually insignificant due to the short distances involved. The coefficient
of contraction Cc is determined through experimental measurements or two-dimensional
analysis of the curvilinear zone. For the simplifying assumption where the approach velocity
is small and therefore, V22/2g = 0, the equation can be further simplified to:
Q = C d hL 2g( y 2 y 1 )
(3.65)
The discharge coefficient Cd is a function of the upstream and downstream depths, the gate
opening, and the gate geometry. The form of the equation as stated above is for both free
and submerged conditions.
3.29
S0 Sf
d( V 2 / 2g)
1+
dy
(3.66)
3.30
Time
Steady Flow
y
Unsteady Flow
y
=0
Space
Uniform Flow
y
=0
Varied Flow
y
Uniform Flow
y
=0
Varied Flow
y
Gradually
Rapidly
Gradually
Rapidly
Varied Flow
Varied Flow
Varied Flow
Varied Flow
Figure 3.22. Flow classification according to change in depth with respect to space
and time.
For a rectangular channel it can be shown that the change of the velocity head with respect
d( V 2 / 2g)
2
is equal to FR and the general differential equation for gradually varied
to depth
dy
flow can be written as:
dy S 0 S f
=
dx
1 FR2
(3.67)
Q2T
.
gA 3
(3.68)
Where ql is the lateral inflow rate per unit length of channel. The momentum equation for
unsteady flow can be written as:
1 V V V
y
+
= S f S0 +
+ ql
g t
g x
x
(3.69)
Equations 3.68 and 3.69 are known as the shallow water equations developed for onedimensional unsteady flow in open channels and are also called the Saint-Venant equations.
Storage Effects for Unsteady Flow. Steady flow computations ignore storage effects both
within the channel and the overbank areas. In comparison, the unsteady flow equations
account for both types of storage. The storage can significantly reduce peak flows giving a
more realistic assessment of the surface flooding. If storage does occur in a floodplain and
flow is treated as steady, then the storage can be accounted for by blocking that portion of
channel by defining it to be ineffective.
3.4.2 Subcritical Versus Supercritical Flow
Various types of waves and surges may occur in open channels and cause a locally
unsteady flow. The simplest is the small surface wave which progresses radially outward
from a point as a rock would cause if thrown into a lake. The rate that this wave progresses
outward is called its celerity. Subcritical flow is when the flow velocity is less than the celerity
of a gravity wave, c = gy , and supercritical flow is when the flow velocity is greater than the
wave velocity.
3.32
c = gy
a) V = 0
b) V < gy
c ) V > gy
Figure 3.23. Propagation of a water wave in shallow water illustrating subcritical and
supercritical flow.
Dropping a rock in a pond will cause a wave to propagate in all directions at the same
velocity (Figure 3.23a). By dropping the rock in a stream and superimposing an average
velocity V of the stream such that V < gy (i.e., V / gy < 1) , the wave will propagate
upstream at a velocity of gy V and downstream at gy + V (Figure 3.23b). This condition
is defined as subcritical flow.
If a higher velocity is imposed such that
V > gy (i.e., V / gy > 1) , the wave will be washed downstream with no affect upstream
(Figure 3.23c). This condition is defined as supercritical flow. The conclusion is that for
subcritical flow any disturbance in the flow field will translate upstream (i.e., water surface
computations must progress from downstream to upstream). On the other hand, for
supercritical flow the computations must progress from upstream to downstream since any
disturbance in the flow field will not translate upstream. In 1861 William Froude presented a
paper where he defined the ratio of the characteristic velocity (average) V to a gravitational
wave velocity c = gy , which was later called the Froude Number.
FR =
(3.70)
gy
Specific Energy. Specific energy is defined as the energy per unit mass. Many practical
problems of open channel flow are solved by application of the energy principle (Bernoulli's
equation) using the channel bottom as the datum as shown in Figure 3.24.
V2/2g
dy
3.33
The concept of specific energy was first introduced by Bakhmeteff in 1912. The water
surface is the same as the hydraulic grade line and the total energy head is represented by
the energy grade line. Specific energy is a fundamental concept and widely utilized in
solving problems of open channel flow. Flow represented in Figure 3.24 is essentially
rectilinear (i.e., the flow has smooth parallel streamlines). The specific energy E is given by
the total energy head consisting of the depth of flow and the velocity head and the total flow
is given by the continuity equation Q = AV. Considering the special case where the head
losses are negligible (hl = 0), the Bernoulli equation can be applied between any two sections
as:
V22
V2
+ y 2 = 1 + y1
2g
2g
(3.71)
The sum of the depth of flow and the velocity head is the specific energy and can be written
as:
E=y+
V2
2g
(3.72)
Since the flow in channel cross section (Figure 3.24) is the product of the area and the
average velocity, a change in the depth will result in a change in the Q unless the velocity
changes inversely to keep the discharge the same. However, the redistribution of the
velocity and the depth keeping the same discharge will result in different specific energies.
Thus there are limitless combinations of velocity and depth that can have the same
discharge. For very high depths the velocity would become very small and the specific
energy would become equal to the depth. On the other hand if the velocity were to become
very high then the depth would become very shallow and the specific energy would be
approximately equal to the velocity head. These trends can be shown in the specific energy
diagram shown in Figure 3.25.
It is obvious from the specific energy diagram that there are two depths for the same energy
except at minimum energy where there is a unique depth. The two depths are referred to as
alternate depths and the depth at minimum energy is typically called critical depth. Critical
depth can be solved by taking the derivative of the energy with respect to depth at the
minimum energy, which in this case will be equal to zero.
dE
d
V2 d
d Q2
y +
=
=
( y) +
dy dy
2g dy
dy A 2 2g
(3.73)
dE
Q2 d 1
= 0 = 1+
dy
2g dy A 2
(3.74)
1+
Q2 d
Q2
1 dA
( A 2 ) = 1 +
( 2) 3
2g dy
2g
A dy
(3.75)
3.34
Minimum E
Alternate Depths
Critical Flow
yc
y
V2/2g
Supercritical Flow
E
Specific Energy = y +
V2
/2g
Q2T
gA 3
(3.76)
Since both the area and the topwidth are functions of depth, Equation 3.76 defines a critical
depth occurring at minimum energy. Substituting V2 = Q2/A2 (i.e., continuity equation Q =
AV), Equation 3.76 can be written in terms of the velocity and a hydraulic depth as:
V 2T
V2
= 1 = FR2 =
gA
gDM
(3.77)
Area
= A/T.
Topwidth
For a rectangular channel the hydraulic depth is equal to the depth of flow and by taking the
square root the equation reduces to the Froude Number introduced at the beginning of this
section (Equation 3.70).
V
FR =
gy
3.35
Dimensional Analysis and Similitude. Many open channel flow problems can be solved only
approximately by analytical or numerical methods. Physical model studies play an important
role in verifying solutions or by providing results that can't be obtained through analytical
solutions.
Similitude is a relationship between a full-scale flow and a flow involving smaller but
geometrically similar boundaries. In 1861 Froude published a paper which dealt with
identifying the most efficient hull shapes. He established a dimensionless number, later to
be called the Froude Number that permitted small-scale tests to predict the behavior of full
sized prototypes.
The three types of similitude involved in fluid mechanics are geometric, kinematic and
dynamic similarity. Geometric similarity involves the length scales, x, y, and z. Ideally the
ratios of the model geometry scales would all be the same. Kinematic similarity involves
length and time ratios, which require that the streamline patterns be the same in the model
as in the prototype. Dynamic similarity requires that the force ratios of model to prototype be
the same from point to point.
F1M F2M F3M
F
=
=
= = nM
F1P F2P F3P
FnP
(3.78)
Typical forces that are encountered in open channel flow are inertia, gravity and viscous. If
the model were to be in absolute similitude (holding to geometric, kinematic and dynamic
similarity) then the model would be the same size as the prototype.
For model studies involving fluid motion, the inertial force has to be included for dynamic
similarity. For open channel flow problems the other dominating force is that of gravity (i.e.,
viscous, surface tension and other forces are small and can be neglected). This requires
that the dynamic force ratio of inertia to gravity for the model must equal that of the
prototype.
ForceInertia
ForceInertia
=
Force
Force
Gravity Model
Gravity Pr ototype
In
(3.79)
terms
be
represented
as
Therefore, the ratio of the inertia force to the gravity force for the model and prototype can be
written as:
l3 V 2
3
l g
l3 V 2
= 3
l g
Model
Pr ototype
(3.80)
3.36
V2
gl
V2
=
Model gl Pr ototype
(3.81)
The length scale can be represented by the depth of flow in many open channel flow
situations. The square root of Equation 3.81 shows that the Froude Number of the model
needs to be equal to the Froude Number of the prototype for open channel similitude. This is
essentially what Froude found when he was performing his experiments in the 1800s.
= V = (FR )M = (FR )P
gy
M gy P
(3.82)
3.37
V2
+y+z
2g
(3.83)
x
hl
V22 2g
V12 2g
y2
H
y1
z2
z1
Datum
3.38
Differentiating Equation 3.83 with respect to x, the rate of energy change is:
dH
d V 2 dy dz
+
+
=
dx dx 2g dx dx
(3.84)
The energy grade line Sf = dH/dL dH/dx for small slopes (i.e., less than 10%). The
channel slope is given by So = dz/dx and the slope of the water surface (hydraulic grade
line) is Sw = dH/dx dy/dx.
The energy equation can be written between sections 2 and 1 for steady flow as:
V22
V2
+ y 2 + z 2 = 1 + y 1 + z 1 + hl
2g
2g
(3.85)
Note that S0 (x) = z2 z1 and Sf (x) hl for small slopes. The energy equation can then be
written as:
y 2 y1 +
V22 V12
= (S f S 0 ) ( x )
2g 2g
(3.86)
Dividing by x and noting in the limit x = dx Equation 3.86 is written in differential form as:
dy
d V2
= S0 S f
+
dx dx 2g
(3.87)
d V2
(3.88)
For gradually varied flow there is a gradual change in depth and velocity with distance and
the conservation of energy with the losses being determined empirically using the Manning
equation. Thus Sf is determined from the Manning equation.
Sf =
Q 2n 2
(1.486) 2 A 2R 4 / 3
(3.89)
Note that the coefficient 1.486 is 1.0 for SI units. For computations of the water surface
elevations along the x-axis a finite difference scheme is used rather than solving the
differential Equation 3.88. Equation 3.85 is written in terms of the water surface WSEL = y +
z as:
WSEL 2 = WSEL 1 +
V12 V22
+ hl
2g 2g
(3.90)
3.39
Equation 3.90 is solved iteratively and is presented in more detail in Section 5.2.
Rapidly Varied Flow. Rapidly varied flow has pronounced curvature of the streamlines;
therefore the hydrostatic pressure distribution assumption is no longer valid. Often the flow
curvature is so abrupt that the flow profile is broken and becomes discontinuous (such as a
hydraulic jump). The energy equation is insufficient to solve for rapidly varied flow problems.
Typically the momentum equation or hydraulic structure relationships are used. Sections
3.3.3 and 3.3.4 presented the weir, gates, and orifice equations that are used to solve for
many rapidly varied flow situations.
Hydraulic Jump. The most important local rapidly varied flow problem is that of the hydraulic
jump, which develops when supercritical flow transitions to subcritical. A hydraulic jump
occurs when high velocity discharges flow into a zone of lower velocity either in natural rivers
or over spillways or other hydraulic structures. Figure 3.28 is an illustration of the hydraulic
jump phenomenon.
The equation relating the depths of flow up and downstream of the jump (y2 and y1) is
developed by applying the conservation of linear momentum to the control volume assuming
the channel bottom has a horizontal or small slope as shown in Figure 3.29. For channels
with small slopes the gravity component (weight of water) in the downstream direction is
relatively small and can be neglected. Also, since the length of the channel is short, the
friction forces are small in comparison to the energy losses through the jump and can be
neglected. Therefore the only significant forces are those caused by hydrostatic pressure
(Figure 3.29).
EGL
hl
y2 < yc
EGL
Flow
yc
y1 > yc
F1 (hydrostatic)
F2 (hydrostatic)
x
Figure 3.29. Control volume for the hydraulic jump.
3.40
By assuming hydrostatic pressure distribution at sections 2 and 1 where the flow streamlines
are nearly parallel and summing the forces in the x-direction gives:
= F2H F1H = m a
(3.91)
The magnitude of the hydrostatic force is F = y A, where y is the distance measured from
the free surface to the center of gravity of the cross sectional area. Therefore Equation 3.91
can be written as:
y2
y
A 2 1 A 1 = m a = Q ( V1 V2 )
2
2
(3.92)
In the case of a rectangular channel A = yW and q = Q/W, Equation 3.92 can be simplified to
by dividing by W:
y 22 y 12 q
= ( V1 V2 )
2
2
g
(3.93)
=
2
2
g y 1 y 2
(3.94)
The next series of equations are algebraic manipulations of the Equation 3.94:
(y 2 y1 ) (y 2 + y1 ) =
y 1y 22 + y 12 y 2 =
2q 2
(y 2 y1 )
gy1y 2
(3.95)
2q 2
g
(3.96)
(3.97)
y 2 y 2 2q 2
+ = 3
y 1 y 1 gy1
(3.98)
This is a quadratic equation for y2 /y1 and knowing y1, can be solved as:
y 2 1
8q 2
= 1 1 + 3
y 1 2
gy1
(3.99)
3.41
This can be reduced to the positive root as the negative root will give a negative depth.
y2 =
1
y1 { 1 + 1 + 8FR2 }
2
(3.100)
(3.101)
Types of Channel Slopes. Equation 3.101 is used to define the type of slopes relative to the
x direction (direction of flow). Since hydrostatic pressure distribution has been assumed in
the development of the equation, the application will be limited to flows with streamlines
essentially straight and parallel, and of small slope. Also, the depth of flow is measured
vertically from the channel bottom, the slope of the water surface dy/dx is relative to this
channel bottom. Figure 3.30 shows water surface slopes resulting from the change in depth
along the channel.
3.42
Vertical
dy/dx =
dy/dx < 0
Horizontal
dy/dx = S0
dy/dx = 0
dy/dx > 0
y
x
Figure 3.30. Water surface slopes versus channel bottom slope.
There are five types of bed slopes that are encountered: mild; critical; steep; horizontal; and
adverse. All of the slopes are defined using the channel bottom slope S0. To define mild,
critical, and steep slopes the channel slope S0 is compared to the slope if the channel were
flowing at critical depth (i.e., solve for the critical slope Sc by substituting yc into the Manning
equation). Then we can define the slopes as follows:
Mild Slope
Critical Slope
Steep Slope
Horizontal Slope
Adverse Slope
S0 < Sc
S0 = Sc
S0 > Sc
S0 = 0
S0 < 0
Figure 3.31 illustrates the 12 flow profile curves for the five slopes. A summary of the profiles
is given in Table 3.4.
When there is a change in cross section geometry or channel slope, or there is an
obstruction to the flow, the qualitative analysis of the flow profile depends on locating the
control points, determining the type of curve upstream and downstream of the control points,
and then sketching the backwater curves. When the flow is supercritical (FR > 1) the control
of the depth is upstream and the computations must proceed in the downstream direction.
On the other hand, when the flow is subcritical (FR < 1) the depth control is downstream and
the computations must proceed upstream. Example flow profile curves that result from a
change in slope are illustrated in Figure 3.32. For the M1, M2, S1, C1, H2, and A2 profiles the
computations for water surface profiles start at the downstream control point and proceed
upstream. For the M3, S2, S3, C3, H3, and A3 profiles the computations for water surface
profiles start upstream at the control point and proceed downstream.
3.43
M1
yn
M2
yc
M3
y > yn > yc
yn > y > yc
yn > yc > y
S1
yn
S2
yc
S3
y > yc > yn
yc > y > yn
yc > yn > y
A2
A3
yc
y > yc
yc > y
3.44
C1
yn
C3
yc
y > yn = yc
yn = yc > y
yc
yn > y > yc
yn > yc > y
Designation
M1
M2
M3
C1
None
C3
S1
S2
S3
None
H2
H3
None
A2
A3
Relation of y to yn and yc
y > yn > yc
yn > y > yc
yn > yc > y
y > yc = yn
y = yc = yn
yc = yn > y
y > yc > yn
yc > y> yn
yc > yn > y
y > yn > yc
yn > y > yc
yn > yc > y
Not applicable
y > yc
yc > y
3.45
Type of Flow
Subcritical
Subcritical
Supercritical
Subcritical
None
Supercritical
Subcritical
Supercritical
Supercritical
None
Subcritical
Supercritical
None
Subcritical
Supercritical
Jump
M1
Jump
S1
M3
Mild Slope
Steep Slope
Milder Slope
Mild Slope
M2
C3
Mild Slope
Steep Slope
Steeper Mild Slope
Critical Slope
M2
S2
S3
Steep Slope
Mild Slope
Steep Slope
C1
Critical Slope
Mild Slope
Mild Slope
Adverse Slope
A2
S2
S2
Critical Slope
Steep Slope
Steep Slope
Adverse Slope
Figure 3.32. Example flow profiles for gradually varied flow with a change in slope.
3.46
CHAPTER 4
HYDRAULIC ANALYSIS CONSIDERATIONS
4.1 INTRODUCTION
Chapter 3 provides background on the fundamental open channel flow concepts that
comprise the basis for the majority of the numerical hydraulic modeling and calculations
encountered in open channel flow and bridge hydraulic analysis. The calculations are often
complex and tedious, and many require iterative solution techniques due to interaction
between variables. Therefore, computer programs have been the primary tool for hydraulic
engineers ever since computers have become widely available. As computer technology has
advanced, so has numerical hydraulic modeling. The primary analysis approach for bridge
hydraulics is one-dimensional modeling, although two-dimensional modeling is becoming
common and three-dimensional modeling is used to analyze complex flow fields. Chapters 5
and 6 provide information and guidance on the use of one- and two-dimensional numeric
models for bridge hydraulic analysis. This chapter includes information on selecting the most
appropriate approach whether it is one-, two-, or three-dimensional numerical modeling,
steady or unsteady modeling, or physical hydraulic modeling. This chapter also provides
background on developing input data and other considerations that are common to all bridge
hydraulic problems regardless of the specific approach.
4.2 HYDRAULIC MODELING CRITERIA AND SELECTION
Any hydraulic model, whether it is numerical or physical, has assumptions and requirements.
It is important for the hydraulic engineer to be aware of and understand the assumptions
because they form the limitations of that approach. It is the goal of any hydraulic model
study to accurately simulate the actual flow condition. Violating the assumptions and
ignoring the limitations will result in a poor representation of the actual hydraulic condition.
Treating the model as a black box will often produce inaccurate results. This is not
acceptable given the cost of bridges and the potential consequences of failure. Therefore,
the approach should be selected based primarily on its advantages and limitations, though
also considering the importance of the structure, potential project impacts, cost, and
schedule.
4.2.1 One-Dimensional Versus Two-Dimensional Modeling
One-dimensional modeling requires that variables (velocity, depth, etc.) change
predominantly in one defined direction, x, along the channel. Because channels are rarely
straight, the computational direction is along the channel centerline. Two-dimensional
models compute the horizontal velocity components (Vx and Vy) or, alternatively, velocity
vector magnitude and direction throughout the model domain. Therefore, two-dimensional
models avoid many assumptions required by one-dimensional models, especially for the
natural, compound channels (free-surface bridge flow channel with floodplains) that make up
the vast majority of bridge crossings over water. Chapters 5 and 6 include detailed
discussions of one- and two-dimensional model assumptions and limitations.
The advantages of two-dimensional modeling include a significant improvement in
calculating hydraulic variables at bridges. Therefore FHWA has a strong preference for the
use of two-dimensional models over one-dimensional models for complex waterway and/or
complex bridge hydraulic analyses. One-dimensional models are best suited for in-channel
flows and when floodplain flows are minor. They are also frequently applicable to small
streams. For extreme flood conditions, one-dimensional models generally provide accurate
results for narrow to moderate floodplain widths. They can also be used for wide floodplains
4.1
when the degree of bridge constriction is small and the floodplain vegetation is not highly
variable. In general, where lateral velocities are small one-dimensional models provide
reasonable results. Avoiding significant lateral velocities is the reason why cross section
placement and orientation are so important for one-dimensional modeling. Two-dimensional
models generally provide more accurate representations of:
Flow distribution
Velocity distribution
Water Surface Elevation
Backwater
Velocity magnitude
Velocity direction
Flow depth
Shear stress
Although this list is general, these variables are essential information for new bridge design,
evaluating existing bridges for scour potential, and countermeasure design. The Federal
Emergency Management Agency (FEMA) also depends on numerical hydraulic models of
extreme events to determine flood hazards. FEMA and NOAA (National Oceanic and
Atmospheric Administration) commissioned the National Research Council (NRC 2009) to
investigate the factors that affect flood map accuracy and identify ways of improving flood
mapping. Among their findings, the NRC recommended greater use of two-dimensional
models.
Two-dimensional models should be used when flow patterns are complex and onedimensional model assumptions are significantly violated. If the hydraulic engineer has great
difficulty in visualizing the flow patterns and setting up a one-dimensional model that
realistically represents the flow field, then two-dimensional modeling should be used. One
study that developed criteria for selecting one- versus two-dimensional models is "Criteria for
Selecting Hydraulic Models" (NCHRP 2006). The recommendations from that study are
summarized and expanded on below.
Multiple Openings. Multiple openings along an embankment are often used on rivers with
wide floodplains. Rather than using a single bridge, additional floodplain bridges are
included. Although one-dimensional models can be configured to analyze multiple openings,
the judgment and assumptions that are made by the hydraulic engineer in combination with
the assumptions and limitations of the software result in an extreme degree of uncertainty in
the results. The proportion of flow going through a particular bridge and the corresponding
flow depth and velocity are important for structure design and scour analysis. Because
multiple opening bridges represent a large investment, two-dimensional analysis is always
warranted.
Another type of multiple opening is multiple bridges in series. There are conditions when this
bridge configuration should be analyzed using two-dimensional models. These include
unmatched bridge openings or foundations that do not align. An upstream or downstream
railroad or parallel road may significantly alter the flow conditions and warrant twodimensional analysis.
Figure 4.1 shows two-dimensional model results (velocity magnitude) for the U.S. Route 1
crossing over the Pee Dee River in South Carolina. Flow is generally from top to bottom in
this figure. This model illustrates several reasons for selecting two-dimensional modeling.
The floodplain width ranges from 4,000 to 8,000 ft (1,200 to 2,400 m) and has highly variable
land use and vegetation. The US 1 crossing includes a 2,000 ft (600 m) main channel bridge
and two 500 ft (150 m) relief bridges. There is also a railroad crossing downstream.
Although the railroad also has three bridge openings, they are shorter and not aligned with
the US 1 bridges. The highest velocity, greater than 8 ft/s (2.4 m/s) occurs in the main
4.2
channel. However, the center relief bridge has an average velocity of nearly 6 ft/s (1.8 m/s)
and the eastern relief bridge has velocities of over 7 ft/s (2.1 m/s). The floodplain area under
the main channel bridge, however, has velocities ranging from 1 to 3.5 ft/s (0.3 to 1.1 m/s).
Therefore, overall conveyance would be improved and backwater would be reduced by
shortening the main channel bridge and lengthening the relief bridges. If changing the bridge
lengths would adversely impact the downstream railroad bridges, the two-dimensional model
results would also quantify those impacts.
Velocity,
ft/s (m/s)
9 (2.7)
8 (2.4)
7 (2.1)
6 (1.8)
5 (1.5)
4 (1.2)
3 (0.9)
2 (0.6)
1 (0.3)
0 (0.0)
4.3
defined, but potential problems with backwater will also be evident. Figure 4.2 shows a
crossing with an approximate 25 degree skew to the floodplain with flow from top to bottom.
This figure illustrates how floodplain impacts can vary greatly upstream of a skewed
crossing. The colors represent the difference in water surface between natural (no bridge
crossing) and existing conditions. The darkest color shows the greatest water surface
increase and the opposite side of the embankment shows a decrease in water surface
compared to natural conditions. The fact that this is also a multiple opening crossing also
complicates the hydraulic conditions.
Increase
Increase
Decrease
No Change
Multiple Channels. Anabranched and braided rivers have multiple channels and flow paths
that complicate hydraulic calculations. Figure 4.4 shows an extreme example of multiple
channels at Altamaha Sound in Georgia. The figure depicts channels in blue, flood-prone
areas in green, and roadway alignments in red. The area is subject to riverine and tidal
flooding. Not only are there nine crossings (five on I-95 and four on SR 17), but there are
more than 20 individual channel segments, or reaches that would need to be included in a
HEC-RAS split-flow model. The hydraulic engineer would also have to decide the amount of
adjacent floodplain to assign to each channel segment and may well need to allow for lateral
flow between floodplain segments. Two-dimensional models, while still a significant
challenge, clearly have numerous advantages in this situation. Although many multiple
channel situations are well simulated with the split-flow options in HEC-RAS, the effort in
developing a two-dimensional model for these conditions may be less than an equivalent
one-dimensional model.
4.6
Embankment
Embankment
Flow Distribution at Bridges. The HEC-18 manual (FHWA 2012b) establishes scour
evaluation procedures recommended by FHWA. Flow and velocity distributions are required
within the bridge opening to calculate contraction, pier and abutment scour. Onedimensional models estimate flow and velocity distribution based on the incremental
conveyance within a cross section (see Section 5.4). This assumption requires that each
point in the cross section also have the same water surface elevation and energy slope.
Figure 4.5 shows water surface and velocity vectors from a two-dimensional model. The
model represents a relatively simple situation, but does not meet the one-dimensional criteria
described above. In this figure, the thin lines indicate the channel banks and embankment.
The water surface is relatively uniform along the upstream face of the bridge, varying by less
than 0.3 ft (0.1 m), but the velocity vectors in the overbank areas in the bridge opening are
not perpendicular to the bridge face. Although these are indicators that the flow is not onedimensional, the most significant departure from one-dimensional assumptions is the velocity
in the overbank areas under the bridge. A one-dimensional model would estimate much
lower velocity in the overbanks based on conveyance and equal energy slope at the bridge
cross section. The average energy slope in the overbank areas under the bridge is over five
times the energy slope of the channel area, resulting in velocities more than twice what is
computed from one-dimensional conveyance-weighted calculations.
Small streams
In-channel flows
Narrow to moderate-width floodplains
Wide floodplains
Minor floodplain constriction
Highly variable floodplain roughness
Highly sinuous channels
Multiple embankment openings
Unmatched multiple openings in series
Low skew roadway alignment (<20)
Moderately skewed roadway alignment (>20 and <30)
Highly skewed roadway alignment (>30 )
Detailed analysis of bends, confluences and angle of attack
Multiple channels
Small tidal streams and rivers
Large tidal waterways and wind-influenced conditions
Detailed flow distribution at bridges
Significant roadway overtopping
Upstream controls
Countermeasure design
/
/
4.8
4.9
(a)
(b)
Figure 4.6. Three-dimensional CFD modeling (a) flow prior to scour
(b) flow at ultimate scour.
Turbulence modeling is an important part of detailed three-dimensional CFD. It takes into
account the fluctuation of velocity and energy transfer/dissipation in the simulation. The
turbulence condition has a significant impact to bridge hydraulics and stream bed stability.
Two important turbulent modeling approaches that are widely applied in bridge-related CFD
simulations are large eddy simulation (LES) and Reynolds-averaged Navier-Stokes (RANS).
The RANS method uses time-averaged equations of motion for fluid. Using Reynolds
decomposition, the instantaneous velocity and pressure fields are decomposed into mean
values and fluctuating components. An additional term compared to the original NavierStokes equations is a tensor quantity, known as the Reynolds stress tensor, in the resulting
equations for the mean quantities. Reynolds stress tensor is modeled in terms of the mean
flow quantities to provide closure of the governing equations.
Explicitly simulating eddies in all scales is extremely demanding on computer power and
impractical. In LES, the turbulence over large scales is resolved by using filtered NavierStokes equations, which is a spatial averaging that eliminates the small scale turbulence.
The small scale eddies are modeled based on the hypothesis that the smaller eddies are
self-similar.
LES allows the computation of instantaneous velocity distribution and
hydrodynamic force, but requires a large amount of computational resources. Because of
desire for information on temporal fluctuation in flow and because of continued
advancements in computer power, the use of LES has increased rapidly in recent years. In
bridge engineering, it is of great interest in scour development because the fluctuation of
hydrodynamic force can significantly increase the erosion potential. In some past studies, it
was found that the high fluctuation of bed shear may occur at a different location than high
mean bed shear (see Figure 4.7).
LES can potentially provide additional temporal details to supplement RANS simulation and
obtain more accurate dynamic measurements. There is not a one-size-fits-all optimal
solution for turbulence modeling, so support from experiments is often needed. Once
numerical modeling is calibrated by experiments, a large amount of additional conditions can
be analyzed and expensive and time-consuming physical modeling can be greatly reduced.
4.10
(a)
(b)
Figure 4.7. Bed shear from LES prior to scour (a) mean bed shear
(b) fluctuation of bed shear.
4.2.4 Physical Modeling
Physical hydraulic modeling has and continues to be a valuable tool in fluid mechanics.
Laboratory scale models provide direct experimental data for complex flow fields, flowstructure interaction, and erosion processes. Fluid mechanics textbooks (such as Munson et
al. 2010) provide in-depth discussion of dimensional analysis and similitude requirements for
laboratory scale models. Geometric similarity is the first requirement, although some
conditions can be evaluated with distorted vertical and horizontal scales. For free-surface
flow conditions, Froude number scaling (the ratio of inertial force to gravitational force)
replicates the dominant hydraulic forces. When the Froude number is used for scaling, other
force ratios, such as the Reynolds number, do not scale. Therefore, physical scale models
are not a complete representation of the prototype conditions. Scale models range from
three-dimensional fixed-bed models to fully three-dimensional moveable-bed models and
moveable-bed models of individual piers or other structural elements to evaluate local scour
(TAC 2004). For moveable-bed models, the sediment characteristics should also be scaled,
though this is often difficult. Figure 4.8 shows a moveable-bed physical model of the I-90
crossing of Schoharie Creek in New York conducted at Colorado State University (CSU).
The model was used to evaluate scour that caused the bridge to fail in 1987. ASCE (2008)
provides a useful discussion of sediment transport scaling for physical models.
4.11
Figure 4.8. Physical model of the I-90 Bridge over Schoharie Creek, New York.
4.3 SELECTING UPSTREAM AND DOWNSTREAM MODEL EXTENT
The minimum extent of a hydraulic model for bridge hydraulics is the location where flow is
fully expanded both upstream and downstream of the flow constriction. Flow constriction is
often the major contributor to backwater, so complete flow expansion and contraction must
be included. For one-dimensional models, the use of the minimum downstream extent does
not detract from the results as long as the downstream water surface is known with a high
degree of certainty. However, if the water surface is not know with confidence, then
extending the model further downstream will decrease uncertainty at the structure. This is
illustrated in Figure 4.9, which shows water surface profiles for a simple bridge model. The
three profiles are all for the same discharge with the only difference being the downstream
boundary condition. Each one of the profiles represents a valid solution to the equations of
fluid motion. The downstream boundary is located far enough downstream so the profiles
converge and the 4.0 feet (1.2 m) of initial difference is eliminated before reaching the bridge.
Thus an important principle of numerical modeling is that the farther downstream the model
extends, the smaller the influence the boundary condition will have on the location of interest.
The farther the boundary is from the bridge, the less uncertainty exists at the bridge because
channel and floodplain geometry and roughness will dictate the results. This does not mean,
however, that all uncertainty is removed. Inaccuracies or change in any of the input
variables result in uncertainty in the results.
The minimum downstream extent for two-dimensional models is similar to one-dimensional
models with flow fully expanded upstream and downstream. It is also desirable to select a
location where flow is reasonably one-dimensional, especially at the downstream boundary.
This is because the downstream boundary is usually specified as a constant water surface
elevation along the boundary. One useful approach is to place the upstream and
downstream boundaries at least one floodplain width up- and downstream of the crossing.
As with one-dimensional models, the further the boundary is located away from the crossing
the less influence the boundary condition will exert of the results.
4.12
When there are other structures or hydraulic controls either upstream or downstream that will
influence or can be impacted by the project, then the modeling should be extended to include
these structures. Figure 4.9 shows some backwater created by the crossing. Although the
extent of the model probably captures the maximum water surface increase, extending the
model upstream would be required to fully assess potential upstream impacts.
As indicated in Section 4.2.2, unsteady flow analysis also requires extending the model to
account for storage-routing effects. Unsteady flow modeling of tidal waterways can require
significant effort. Tidal models must extend far enough downstream to reach a well-defined
tide or storm surge boundary condition and to account for storage and hydraulic controls
between the downstream boundary and the structure. Tidal models must extend far enough
upstream of the structure to account for storage because it is the storage that is the primary
factor that determines tidal flow rates (FHWA 2004, 2008).
1020
W
Legend
Water Surface Profiles
Elevation
1010
Ground
1000
990
980
1000
2000
3000
4000
Distance
Figure 4.9. Flow profiles with downstream boundary uncertainty.
4.4 IDENTIFYING AND SELECTING MODEL BOUNDARY CONDITIONS
An important part of the hydraulic engineer's responsibility is to select representative
boundary conditions for the hydraulic analysis. Peak discharge is one boundary condition
that is commonly used for river projects and flood hydrographs are most frequently used for
unsteady riverine modeling. For subcritical flow conditions, the downstream water surface
must be specified or computed. For supercritical flow the upstream condition is specified and
for mixed flow conditions the downstream and upstream condition is specified. The model
extent (Section 4.3) and boundary condition should be selected based on identifiable
hydraulic controls or on other reliable information. There are several types of hydraulic
controls that can establish the boundary condition at a bridge. These include slope breaks
4.13
where critical depth occurs (from flat to steep in the downstream direction), diversion dams,
bridges, roads and other structures. The discussion below relates to a downstream
boundary but also applies to upstream boundary conditions for supercritical or mixed flow
models.
4.4.1 Water Surface
A known water surface is very commonly used in hydraulic modeling, where the hydraulic
engineer specifies the elevation as the starting downstream condition. One common source
for the known water surface is a Flood Insurance Study (FIS). The FIS profile may include
many miles of river downstream of the bridge that is being analyzed. Starting the model
relatively close to the bridge is more efficient and the water surface can be extracted from the
appropriate location on the profile. Gage data or an observed high water mark can also be
used to establish known water surface elevations as input boundary conditions.
4.4.2 Normal Depth and Energy Slope
Normal depth occurs when the bed profile, water surface, and energy grade line are all
parallel, and the flow depth and velocity do not change along the channel flow path. This
occurs relatively infrequently in natural rivers, though it can be a reasonable approximation
for establishing boundary conditions in many situations. The use of the channel invert
(thalweg) to compute bed slope should be avoided in natural channels because the channel
bed elevation can vary widely over short distances. A better approximation is to use the
floodplain slope as measured from a topographic map. The channel slope can, however, be
much less than the valley slope for highly sinuous meandering channels. A conveyance
weighted slope can be used, but this requires an assumed water surface to compute channel
and floodplain conveyance.
When energy slope or normal depth is used, the model iteratively computes a water surface
that produces the desired slope. Flow conditions are unlikely to actually satisfy normal depth
criteria because of longitudinal topographic and roughness variations.
The model
downstream extent should be extended for this situation. The variability in channel and
floodplain conditions is then incorporated into the model solution and uncertainty caused by
the boundary condition is reduced.
There are situations when the FEMA Flood Insurance Study (FIS) only includes the 100-year
flood profile but the bridge hydraulic study requires additional design flows. Similarly, the
FEMA study may include several flood discharges but these do not match those desired for
the hydraulic study. For this condition the energy slope can be computed from one
discharge and applied to other discharges. This approach may also involve significant
uncertainty and should be used in conjunction with extending the modeling downstream.
4.4.3 Rating Curve
A rating curve is a stage versus discharge table or curve relating stage and discharge.
Gaging stations have published rating curves that are regularly checked and updated by the
USGS. Gaging station data can also be used to establish rating curves. These data only
apply to the specific gage location. Multiple profile data from FEMA studies can also be used
to develop rating curves for a specific location and used as a model boundary condition. The
same uncertainties can apply to the use of energy slope, so extending the model
downstream is warranted.
4.14
The USGS StreamStats web-based application computes stream flow statistics for gaged
and ungaged locations throughout the U.S. (USGS 2008). For states that do not have
StreamStats fully implemented, USGS gaging station statistics are provided. States that
have StreamStats implemented include gage analysis, gage transfer, and application of
regional regression equations.
4.6 NUMERICAL MODEL EVALUATION
Numerical model verification, calibration and validation are all part of the evaluation process.
Schwartz (2005) indicates that model verification involves testing to assure that the model
solve the equations correctly. The verification process may include testing the model results
against known analytical solutions to the same set of conditions. Although it can be
assumed that widely used and accepted one- and two-dimensional models solve the
4.15
appropriate equations correctly, errors in the programs do become evident from time to time.
Therefore, it is the hydraulic engineer's responsibility to check results for reasonableness.
Even though a program is correctly solving the equations, errors in data entry should also be
checked.
Even though a model has been verified and all the input data are correct, it can produce
erroneous results. This can occur in one-dimensional models if cross section spacing is too
large and in two-dimensional models if the network is not sufficiently refined to solve the
equations accurately. This type of error can be identified by reviewing model results. As
discussed in Section 4.7, inaccurate or incorrect data of one type, particularly elevation, may
require the use of unrealistic values of other parameters, such as roughness, to compensate.
A solution to a specific set of conditions also requires appropriate boundary conditions and,
in the case of unsteady flow models, appropriate initial conditions. For a set of boundary and
initial conditions, the model parameters (including roughness, turbulence, and other
coefficients) must be adjusted to calibrate the model to match observed conditions within a
desired degree of accuracy. If the calibrated parameters are not within the normal expected
range, the model should be reviewed to determine if there are errors in the input data. In the
case of hydraulic models, errors in geometry are often the source of unrealistic results or the
need for unrealistic input parameters.
If possible, the calibration process should not use all available observed data. Part of the
data, especially observations from another event, should be reserved for the validation
process. The validation step tests the model and can improve the confidence in the model
results, but may also identify deficiencies in the model.
Schwartz (2005) also includes sensitivity analysis and uncertainty analysis as part of
numerical model evaluation. Sensitivity analysis, where each parameter is adjusted
independently, is used to identify the parameters that have the greatest impact on the
solution. Because numerical hydraulic models are used to simulate complex systems, any
one parameter may dominate the solution. Uncertainty analysis is similar to sensitivity
analysis, but is used to evaluate the overall uncertainty in the model results based on the
uncertainties of the model input parameters. Monte-Carlo methods, which allow a set of
input parameters to vary randomly based on expected probability characteristics, are very
useful in determining modeling uncertainty.
4.7 DATA REQUIREMENTS AND SOURCES
There is a wide variety of information that is pertinent to bridge hydraulics and scour
analyses. Table 4.2 provides a summary of the various types of information, their use, and
sources. Although all of the data listed in Table 4.2 can be important in a bridge hydraulic
study, geometric data are the greatest source of uncertainty and error. If the geometry is
incorrect, then the flow, velocity, depth or water surface elevation must be incorrect. For
example, if the floodplain elevation is several feet low and the modeled water surface is
correct then the flow depth, and probably the velocity and floodplain discharge are incorrect.
To obtain the correct velocity and discharge with an incorrect depth, then some other
variable, probably flow resistance, must be adjusted incorrectly. That variable may well be
outside the normal expected range. For these conditions, a model that has been calibrated
for one flow is unlikely to produce accurate results for another event.
4.16
Use
Hydraulic model geometry
Existing structure
information
FEMA Flood Insurance
Studies and other flood
hazard studies
Flood maps
Core Samples
Floodplain and channel
roughness
Bed and bank sediment
surface and near-surface
samples
Coastal hydrographic survey
maps and data and coastal
DEMs
Channel stability
assessment
Flood frequency analysis,
historic flooding, hydraulic
model calibration and
validation
Astronomic tide, water
surface elevation frequency
analysis
4.17
Sources
Land survey,
photogrammetry, LIDAR,
USGS National Elevation
Dataset (NED)
Land survey, hydrographic
survey, LIDAR
Photogrammetry, web, city,
county, and state agencies
USGS, Farm Service
Agency (FSA), web, city,
county and state agencies
Bridge plans, as-built plans,
roadway plans
Federal Emergency
Management Agency
(FEMA), U.S. Army Corps of
Engineers (USACE), local
floodplain administrator
FEMA and local floodplain
administrator
FEMA, local floodplain
administrator, USACE, other
Geotechnical investigation
Geotechnical investigation
Site visit
Site visit
NOAA
The variable that has the greatest effect on accuracy is topographic data and the need for
increased accuracy of elevation data increases for lower relief areas (NRC 2009). Geometric
accuracy includes elevation, reach lengths, and bridge and roadway geometry. Improved
elevation accuracy improves the results of all models. There is often the misconception that
two-dimensional models require more accurate data and a larger domain. Two-dimensional
models produce better results because they include more complete representations of the
physical processes. If a topographic or vegetation feature needs to be incorporated in a twodimensional model, it should also be incorporated in a one-dimensional model. Therefore,
the complexity of the flow situation is the primary reason for selecting two-dimensional
models, not data accuracy.
4.18
CHAPTER 5
ONE-DIMENSIONAL BRIDGE HYDRAULIC ANALYSIS
5.1 INTRODUCTION
The previous chapter describes many differences between one-dimensional and two- or
three-dimensional hydraulic analysis. As stated, most bridge hydraulic studies employ onedimensional analysis methods. This chapter provides information and guidance on the use of
one-dimensional modeling techniques for bridge hydraulic analysis.
One-dimensional analysis encompasses a wide range of approaches from approximate
methods requiring just a single waterway cross section to detailed water surface profile
calculations involving many cross sections and multiple stream reaches. Approximate
methods are frequently used for rapid assessment of flood inundation potential in support of
FEMA floodplain mapping. They typically incorporate the assumption of uniform flow (see
Chapter 3). If uniform flow is assumed, then the flow depth and corresponding water surface
elevation at a particular cross section can be calculated using the Manning equation
(Equation 3.46). The HDS 1 method described in the next section is an example of an
approximate method. It includes an underlying assumption that flow conditions are
essentially uniform downstream of the bridge, and it develops a backwater estimate using
empirical equations based on energy loss principles.
The engineer must be cautious, however, in applying approximate methods to bridge
hydraulics problems. Bridge-constricted floodplains and stream reaches usually exhibit
significantly non-uniform flow characteristics. It is recommended, therefore, that engineers
use methods employing water surface profile calculations for one-dimensional bridge
hydraulic analysis.
5.2 HDS 1 METHOD
As explained in Chapter 1, the predecessor to this document is HDS 1 (FHWA 1978). HDS 1
presented a computational method of determining the backwater caused by a bridge
crossing a floodplain. Chapter II of HDS 1 presented the basic expression for backwater as:
h1* = K * 2
A
Vn22
+ 1 n 2
2g
A 4
A n2
A1
Vn22
2g
(5.1)
where:
h1*
K*
1, 2
An2
=
=
=
=
Vn2
A1
=
=
A4
The HDS 1 backwater expression (Equation 5.1) applies the energy equation between the
location of maximum backwater upstream of the bridge and the point downstream of the
bridge where flow is fully expanded, and including an empirical bridge loss coefficient. The
expression is based on the assumption of steady, subcritical flow in the affected stream
reach (classified in HDS 1 as Type I flow). Another significant assumption inherent in the
expression is that the flow conditions are approximately uniform (a uniform water surface
slope parallel to the stream bed slope) except for the backwater caused by the bridge. In the
framework of HDS 1, Cross Section 1 is upstream of the bridge at the presumed point of
maximum backwater, Cross Section 2 is at the upstream face of the bridge, and Cross
Section 4 is downstream of the bridge at the presumed point of reestablishment of normal
flow conditions (see Figure 5.1). Cross Section 3 is located at the toe of the downstream
side slope of the road embankment, but it is not used in the calculation of backwater.
Figure 5.1. Sketch illustrating positions of Cross Sections 1 through 4 in HDS 1 backwater
method (FHWA 1978).
5.2
The engineer applying the HDS 1 method would first compute the flow depth in a
representative cross section under uniform flow conditions and without any constriction, for a
given design discharge rate. Prior to incorporating the constriction caused by the bridge
crossing, the representative cross-section properties would apply to each of the cross
sections (1 through 4) because of the uniform flow assumption. Once the unconstricted flow
depth is determined, the engineer computes A4 and 1. The values of An2, Vn2 and 2 are
then computed based on superimposing the constriction caused by the road embankments
and abutments onto the cross-sectional area and considering the area within the constriction
and under the normal water surface (see Part C of Figure 5.1).
The engineer would then determine the bridge opening ratio (M), which represents the
degree of constriction of the waterway. The value of M is computed by dividing the total
cross section discharge (Q) into the discharge (Qb) passing through the bridge without
redirection by the encroaching road embankments or abutments.
M=
Qb
Q
(5.2)
where:
Qb
Discharge that can pass through the bridge without redirection by the
encroaching road embankments or abutments, cfs. Referring to Part D of
Figure 5.1, Qb is computed as the discharge contained in the portion of the
representative unconstricted cross section that lies within the projected
limits of the bridge opening, ft3/s (m3/s)
Total cross section discharge, ft3/s (m3/s)
After computing the value of M, the engineer would develop the value of K*. The value of K*
is found through a series of graphical charts that were derived empirically from physical
modeling. K* is primarily a function of M, but is also affected incrementally by other factors
including the skew angle (if any), the size and type of bridge piers, and the eccentricity of the
bridge opening within the floodplain. Once the value of K* was obtained, the total backwater
height h1* could be computed, which in turn would allow the upstream water surface elevation
to be calculated.
Because the HDS 1 method incorporated the concept of uniform flow, the backwater could
be estimated through relatively straightforward calculations, avoiding the complexity of the
step backwater calculations associated with varied flow. The uniform-flow simplification,
however, meant that the method would yield uncertain results when applied to situations
involving highly varied flow conditions.
In addition to the basic backwater computation method, HDS 1 provided additional methods
for bridges experiencing certain complex flow conditions, including:
Flow passing through critical depth inside the constriction (Type II flow)
Submerged-deck (pressure) flow conditions
Flow overtopping the road
Bridges with spur dikes (now called guide banks)
The methods presented in HDS 1 for submerged-deck flow and overtopping flow are still in
common use at present, and are discussed in later sections of this chapter.
5.3
Flow
5.4
Figure 5.3 graphically illustrates the computational framework for solving the energy equation
using the standard-step approach. The calculations progress along the length of the stream
segment, one cross section at a time. The hydraulic solution from the previously calculated
cross section (Cross Section 1 in Figure 5.3) and the user-specified information about the
cross section currently being calculated (Cross Section 2) are used to calculate the energy
loss between the two cross sections. Once the energy loss has been determined, the energy
grade line and water surface elevation at the current cross section can be computed. While
the specific details of the implementation of the standard-step approach vary depending on
the program used, the basic steps can be generally described as follows:
1. Set a trial water surface elevation for Cross Section 2. In HEC-RAS the initial trial water
surface is determined by assuming the depth is the same as that calculated for Cross
Section 1.
2. Use the trial water surface elevation to compute the conveyance, energy (friction) slope,
kinetic energy distribution coefficient, and velocity head at Cross Section 2 (see Chapter
3).
3. Compute the average friction slope between Cross Sections 1 and 2, and multiply the
average by the reach length between the two cross sections to estimate the friction loss.
he
V22
2g
V12
2g
Y2
Y1
Z2
Z1
Datum
Figure 5.3. Illustration of water surface profile between two cross sections.
4. Multiply the absolute difference in the velocity head values between Cross Sections 1
and 2 by either a contraction loss coefficient or an expansion loss coefficient to estimate
the transition loss.
5. Find the total energy loss (on a trial basis) from Cross Section 1 to Cross Section 2 by
adding the friction loss and the transition loss.
6. Compute the energy grade line elevation (on a trial basis) at Cross Section 2 by adding
the total energy loss to the energy grade line at Cross Section 1.
5.5
7. Subtract the velocity head from the energy grade line at Cross Section 2 to compute the
resulting water surface elevation.
8. Compare the water surface elevation computed in step 7 to the trial water surface
determined in step 1.
9. If the difference is within the specified tolerance, the calculation for Cross Section 2 is
complete and the calculations progress to the next cross section. If the difference
exceeds the tolerance, assign a new, adjusted trial water surface elevation and begin
again at step 2. Iterate until the difference between the computed and trial water surface
elevations is within the specified tolerance, then progress to the next cross section.
The procedure described above operates on two cross sections at a time. The calculation at
any cross section (e.g., Cross Section 2) requires the information from the solution at the
previous cross section (Cross Section 1). The analysis, therefore, must start with a known
water surface elevation at the first cross section in the reach. For subcritical analysis, the
downstream-most cross section is the first cross section. For supercritical analysis the
calculation starts with the upstream-most cross section. This known water surface elevation
at the first cross section is termed the "starting water surface" or the "boundary condition."
Most bridge-related studies involve subcritical flow and therefore require the engineer to
specify a downstream boundary condition. This specified value is typically taken from a prior
study (such as a FEMA Flood Insurance Study profile plot) or is calculated using the
Manning equation, which requires an assumption of uniform flow conditions in the reach
downstream of the analysis. If the downstream end of the model is located at a free overfall
(such as a grade control structure) or a slope change from flat upstream to steep
downstream, it may be appropriate to assign the water surface elevation at critical depth as
the boundary condition.
Step 3 in the description above involves approximating the friction loss using the average
friction slope. The friction slope at any cross section is:
Q
Sf =
K
(5.3)
where:
Sf
K
=
=
The friction loss between two cross sections is the integral of the friction loss function. An
analytical solution would be highly complex. A simplified numerical integration is achieved by
multiplying the average slope by the distance between the cross sections. The average
friction slope between two cross sections is typically computed by one of four methods
(USACE 2010c):
The Average Conveyance Equation
Q + Q2
Sf = 1
K1 + K 2
(5.4)
5.6
(5.5)
Sf S f1 S f 2
(5.6)
2(S f1 x S f 2 )
S f1 S f 2
(5.7)
The friction slope is inversely related to the square of conveyance. In stream reaches with
significant variation in cross section geometry, one can expect a corresponding variation in
the conveyance values, which in turn could lead to large changes in the friction slope from
one cross section to the next (see Figure 5.4). Since the standard step method uses the
average friction slope between two cross sections to compute the friction loss, a large
change in the friction slope can reduce the accuracy of the calculation.
Cross Section 3
Cross Section 2
Cross Section 1
Figure 5.4. Illustration of the friction slope (the slope of the energy grade line) at each cross
section in a stream segment.
5.7
Chapter 3 of this manual describes the various possible profile types on mild and steep
slopes (M1, M2, S1, S2, etc). Each of the four friction slope averaging methods above is
more suitable for some profile types than others. The HEC-RAS Hydraulic Reference Manual
(USACE 2010c) quotes research by Reed and Wolfkill (1976) indicating that the average
conveyance equation (Equation 5.4) gives the best overall results for a wide range of profile
types. The average friction slope equation (Equation 5.5) is the most suitable method for M1
profiles. For M2 profiles the harmonic mean equation (Equation 5.7) was shown to be the
most suitable. Any one of the friction slope averaging methods will produce accurate
results if the reach lengths are sufficiently short (USACE 1986a).
As the reach length increases, however, so does the potential error in the computed average
friction slope, regardless of the selected averaging method. Additionally, as the reach length
increases, the error in the average friction slope is applied over a longer distance, thus
having a greater effect on the total friction loss. The best remedy for the potential inaccuracy
stemming from the variation of friction slope is to keep the reach lengths short from one
cross section to the next.
As stated earlier, a water surface profile model using the standard step method should have
cross sections located at all locations necessary to represent the major transitions in cross
section geometry. Usually, however, additional cross sections are required to keep the reach
lengths short enough to avoid significant error in the average friction slope and total friction
loss calculations (USACE 1986a). The additional cross sections are often inserted using the
interpolation function of the program being used. An advisable practice in developing a
standard-step model is to shorten the reach lengths (e.g., add cross sections) in successive
trials until the resulting water surface profile is insensitive to further shortening of the reach
length. In other words, the number of cross sections is sufficient when inserting more cross
sections does not significantly change the results.
5.3.2 Other Water Surface Profile Methods
Direct Step Method. The direct step method is similar to the Standard Step Method in that it
uses average friction slope between two locations along the channel to compute the friction
loss term in the energy equation. It is also similar in the fact that calculation of the water
surface profile progresses from a known condition at one cross section (1) to another cross
section (2). It is not a trial and error approach because the water surface at second cross
section is determined in advance of the calculation, which is the flow depth at section 1 plus
some increment in flow depth. The primary limitation of the direct step method is that the
channel must be prismatic, meaning no change in the geometry, roughness, or discharge
between the cross sections. Therefore, this approach is not applicable to natural channels.
The steps in the Direct Step method are:
1. Calculate the specific energy (E = y + V2/2g) and energy slope (Equation 5.3) for flow
depth, y1, at cross section 1
2. Based on a change in flow depth (y) and resulting y2, calculate the specific energy and
energy slope at cross section 2.
3. Calculate the average energy slope (Sf) between the two cross sections.
4. From the energy equation, the distance between the two cross sections is:
x =
E
S 0 Sf
(5.8)
5.8
This method is limited to prismatic channels because the channel properties must apply to
any location along the channel.
Integration Methods. The direct and exact integration of the energy equation for all types of
channels and flow conditions is not possible, though many attempts have been made to
solve the equation for specific cases or through the use of simplifying assumptions (Chow
1959, Henderson 1966, Chaudhry 2008 and others). The differential equation explicitly
containing the independent variables (Chaudhry 2008) is:
dy
=
dx
S0 S f
WQ 2
1
3
gA
(5.9)
Each of the variables on the right side of the equation can vary with distance along the
channel. If one assumes that discharge and Manning n are constant along the channel
reach, that conveyance is proportional to yN/2, and that A3/W is proportional to yM, then
Equation 5.9 can be written as:
N
y
1 0
dy
y
= S0
M
dx
yc
1
y
(5.10)
According to Henderson, if the channel is rectangular, very wide, and the Manning equation
is used, then N = 3 and M = 10/3. The assumptions and difficulties in integration make this
method limited to fairly short reach lengths.
Numerical Methods. The Standard and Direct Step Methods represent commonly used
numerical integration of the energy equation. Other numerical approaches include single
step methods, including Euler, Improved Euler, Modified Euler and Fourth-Order RungaKutta, and Predictor-Corrector methods (Chaudhry 2008). The single step methods
represent successive improvements of computing average energy slope between cross
sections. The Predictor-Corrector methods, including the Standard Step Method, involved
iteration to arrive at a water surface computed to within and acceptable tolerance. Any of
these methods is still limited by the fact that the mean (or integrated) energy slope must not
be computed over too great of a distance. Therefore, the approach recommended in HECRAS (USACE 2010c) to limit cross section spacing is both necessary and robust. It should
also be noted that in practical applications, discharge and Manning n will also vary
longitudinally. Also the left floodplain, right floodplain, and channel distances between cross
sections will not be equal. Therefore, many other assumptions are required for solution of
water surface profiles using one-dimensional methods.
5.3.3 Mixed-Flow Regime
Natural streams and floodplains flow predominantly in the subcritical flow regime, and
therefore most bridge hydraulic analyses are concerned exclusively with subcritical flow.
Occasionally, however, supercritical flows are present within segments of the stream reach
being analyzed. A water surface profile model that includes both subcritical and supercritical
flow segments requires a mixed flow regime. The HEC-RAS Hydraulic Reference Manual
5.9
(USACE 2010c) describes the process used by the HEC-RAS program to analyze mixedflow water surface profiles. The process is paraphrased here.
The program starts by computing a subcritical water surface profile, starting at the
downstream most cross section and working in the upstream direction. After completing the
subcritical profile, if the user has indicated that a mixed-flow profile is to be calculated, the
program begins a supercritical profile calculation starting at the upstream end of the model.
Working its way downstream, the program computes a supercritical profile wherever such a
profile is possible. Some cross sections will be found to have valid solutions for both
subcritical and supercritical flow. At such locations, the program determines which solution
controls by computing the specific force for each solution. Whichever solution has the greater
specific force is the controlling solution. The specific force is computed using the following
expression:
SF =
Q 2
+ AtY
gA m
(5.11)
where:
SF
Am
At
=
=
=
=
The program completes the mixed-regime profile after identifying the controlling solution at
each cross section. The upstream end of a hydraulic jump (abrupt transition from
supercritical flow upstream to subcritical flow downstream) can be located through the mixedregime profile calculations.
Figure 5.5 is an example of a mixed-flow water-surface profile computed by HEC-RAS. In
this profile, a mild slope flows into a downstream steep slope, passing through critical depth
at the slope break. Flow is subcritical at both boundaries, but there is an internal segment of
supercritical flow (S2 curve) upstream of the hydraulic jump up to the slope break. There is a
subcritical profile (S1 curve) at the downstream end of the steep slope controlled by a high
water surface elevation at the downstream boundary. Had the downstream boundary been
set as normal depth for the steep slope, the entire steep slope would have been an S2 curve.
There is an M2 profile downstream of the bridge crossing and an M1 profile upstream of the
bridge crossing. Each of these mild-slope profiles converge on normal depth. The various
types of flow profiles are described in Section 3.4.4.
5.4 CROSS-SECTION SUBDIVISION AND INEFFECTIVE FLOW
5.4.1 Cross Section Subdivision
Equation 5.3 shows that the friction slope is inversely related to the square of the
conveyance, K. In a natural floodplain, the flow depth, roughness and velocity usually vary
throughout the width of the cross section. Accurate conveyance calculations usually require
that the cross section be subdivided into regions of similar flow properties as illustrated in
Figure 5.6.
5.10
30
Bridge
Backwater
M1
W
C
25
N
M2
S2
Elevation
S1
20
Legend
Hydraulic Jump
Water Surface
Critical Depth
15
Normal Depth
Ground
10
0
500
1000
1500
2000
2500
Distance
Figure 5.5 Example mixed-flow regime profile in HEC-RAS.
As an illustration of the inaccuracy that could result without cross section subdivision,
consider the cross section in Figure 5.6 (but without the indicated subdivisions) for a
condition in which the water surface is just below the top of the left bank of the main channel.
The conveyance would reflect only the area and hydraulic radius of the main channel. Next
consider a water surface elevation just above the top of the left bank. At this elevation the
water surface would extend out onto the left overbank. If the cross section were not
subdivided, the wetted perimeter might increase by hundreds of feet, but the area would
increase very little because of the small depth of flow on the overbank. The much-increased
wetted perimeter divided into the little-increased area would lead to a decrease in the
hydraulic radius, which would lead to a decrease in the conveyance compared to the water
surface just below the channel bank. This condition causes a discontinuity in the calculated
conveyance as a function of water surface elevation (see Figure 5.7).
In reality, the small increase in water surface elevation would lead to a small increase in
conveyance, but in a conveyance calculation without subdivision, the conveyance would
appear to decrease in response to the increased water surface. This is an error that can be
avoided through cross section subdivision.
The key concept to consider in subdividing a cross section is that the velocity inside any
subdivision should be more or less uniform, even if the average velocities in two adjacent
subdivisions are significantly different. The total conveyance of a cross section is the
summation of the conveyance for each subdivision of the cross section.
K=
K
N
(5.12)
where:
Ki
N
5.11
Main channel
Deep flow,
High velocity
Left overbank
Shallow flow, low velocity
K1
K2
Right overbank
Shallow flow, low velocity
K3
K4
5.12
2
1.49
A iR i 3
ni
(5.13)
where:
ni
Ai
Ri
=
=
=
In the example shown in Figure 5.6, the cross section has four subdivisions, two in the left
overbank and one each in the main channel and the right overbank. It is recommended to
subdivide overbank areas at changes in roughness. In the main channel, however, it is more
common to treat variable roughness (e.g., willows on the upper channel banks with an
unvegetated channel bottom) by calculating a single composite Manning roughness value
that applies to the entire channel width.
The subdivision of conveyance, in addition to refining the accuracy of the total cross section
conveyance, also provides a rational means of distributing the total discharge within the
cross section. Most one-dimensional analysis programs, including HEC-RAS, distribute
discharge within a cross section such that it is proportional to the conveyance. If the left
overbank has one sixth of the total conveyance, for instance, the program will assign it one
sixth of the total discharge. Distributing the discharge is important in the calculation of the
velocity distribution coefficient, the representative reach length between cross sections, and
the average velocity in each subdivision. An approximate determination of the lateral velocity
distribution is possible by further dividing the main channel and overbank regions into smaller
subdivisions and distributing the discharge to each subdivision in proportion to conveyance.
5.4.2 Ineffective Flow
In a one-dimensional model, the program assumes that any area in a cross section below the
water surface elevation is available for conveyance, and makes use of that conveyance
unless the user specifies otherwise. In certain cases the engineer may want a portion of a
cross section to be excluded from the conveyance, for one of several reasons including:
That portion of the cross section is in the stagnant or eddying wake area downstream of a
large obstruction, such as a building
It is immediately upstream of an obstruction such that any water moving in the area is in
a lateral direction rather than in the downstream direction
It is an area where water can pond but cannot effectively convey flow from upstream to
downstream, such as an area behind a levee that is connected to the flowing water
downstream but not upstream
5.13
Ineffective
flow area
100-year
water surface
Road
embankment
Upstream Bounding
Cross Section
Engineers have used various approaches to exclude portions of cross sections from the
conveyance, including raising the ground elevation value or artificially increasing the
roughness coefficient. In the HEC-RAS program, the user can specify areas within a cross
section where the flow is ineffective (the Ineffective Flow setting) up to a user-designated
water surface elevation. If the water surface in the cross section reaches the designated
elevation, the ineffective flow specification is nullified. The use of ineffective flow
specifications plays an important role in modeling bridge crossings with HEC-RAS, as
discussed in later sections. Figure 5.8 is an example of the use of ineffective flow
specifications at a bridge crossing.
5.14
When flow is constricted by a bridge or culvert crossing, the energy losses in the region
upstream and downstream of the crossing are greater than they would be without the
constriction. The flow upstream of the bridge is forced to contract from the full floodplain
width to the structure opening width. On the downstream side the flow re-expands to occupy
the full floodplain width. Both the contraction and expansion processes require some
longitudinal distance from the crossing for fully established flow. This manual refers to the
zones of establishment as the contraction and expansion reaches, or collectively as the
transition reaches. Increased friction losses (due to decreased conveyance) and transition
losses characterize both the contraction reach and the expansion reach.
5.5.2 Cross Section Placement
In one-dimensional hydraulic modeling, the engineer's placement of cross sections and the
modification of the conveyance properties of certain cross sections drive the analysis of the
transition reaches. Figure 5.9 illustrates the typical one-dimensional modeling framework for
analyzing a bridge crossing. The key concept for modeling transitions is the narrowing of the
effective flow width in the contraction reach and the widening of the effective flow width in the
expansion reach. The outer streamlines in the transition reaches would naturally follow
curvilinear flow paths. In one-dimensional modeling, however, the engineer typically
simplifies the problem by assuming linear transition tapers as shown on Figure 5.9.
The hydraulic model's computation of the excess loss is directly related to the length of the
contraction reach (the distance from the approach section to the upstream bounding section)
and the length of the expansion reach (the distance from the downstream bounding section
to the exit section). The longer the transition reach, the more excess loss is expected. The
contraction and expansion reach lengths are directly determined by the locations of the
approach and exit sections, which are assigned by the engineer. This situation illuminates
one of the limitations of one-dimensional analysis in modeling constrictions. In twodimensional modeling, the model's governing equations determine the length and
configuration of the transition flow regions. In one-dimensional modeling, the engineer
imposes the length and configuration of the transitions on the model through the placement
of the cross sections and the modification of the conveyance properties.
The placement of the approach and exit sections depends on the engineer's assessment of
the appropriate rates of taper for the contracting and expanding flow. The taper rates CR and
ER on Figure 5.9 (contraction rate and expansion rate, respectively) vary depending on
many site-specific factors. Chapter 5 of the HEC-RAS Hydraulic Reference Manual provides
guidance on the assignment of CR and ER (USACE 2010c).
Expansion Reach. Table 5.1 below is taken from the Hydraulic Reference Manual. The table
summarizes the results of research carried out by the USACE Hydrologic Engineering Center
and documented in Research Document 42 (USACE 1995). It gives the ranges of expected
values of ER for different combinations of degree of constriction, longitudinal slope, and ratio
of roughness between the overbanks and main channel. A cell in the table is selected based
on the degree of constriction, slope and roughness ratio that are closest to those of the site
being analyzed. The selected cell gives the range of appropriate ER value.
The engineer decides on a value within the range in the cell and uses that value to estimate
the length of the expansion reach. The expansion reach length is the distance required for
the effective flow to expand to the edges of the floodplain at the ER taper rate. For example if
an ER value of 2 is chosen, and the floodplain encroachment distance is 100 feet (30 m) on
one side of the floodplain, the flow will take 200 feet (60 m) to fully expand on that side. For
asymmetric encroachments, where the constriction is more pronounced on one side of the
floodplain than the other, the expansion reach length can be based on the average
encroachment distance.
5.15
Figure 5.9. Illustration of flow transitions upstream and downstream of a bridge crossing.
Table 5.1. Ranges of Expansion Rates, ER (after USACE 2010c).
b/B ratio
Slope
nob/nc = 1
nob/nc = 2
nob/nc = 4
0.10
0.0002
1.4 3.6
1.3 3.0
1.2 2.1
0.10
0.001
1.0 2.5
0.8 2.0
0.8 2.0
0.10
0.002
1.0 2.2
0.8 2.0
0.8 2.0
0.25
0.0002
1.6 3.0
1.4 2.5
1.2 2.0
0.25
0.001
1.5 2.5
1.3 2.0
1.3 2.0
0.25
0.002
1.5 2.0
1.3 2.0
1.3 2.0
0.50
0.0002
1.4 2.6
1.3 1.9
1.2 1.4
0.50
0.001
1.3 2.1
1.2 1.6
1.0 1.4
0.50
0.002
1.3 2.0
1.2 1.5
1.0 1.4
Variables: b = bridge length, B = expanded flow width, S = slope, nc = channel Manning n,
nob = overbank Manning n.
Once the expansion reach length has been estimated, the engineer locates the exit cross
section and plots it on a topographic map or aerial photograph. Often the expansion reach is
so long that the engineer will want to insert intermediate cross sections between the bridge
and the exit section. Inserting intermediate cross sections is encouraged as long as
ineffective flow specifications are used to represent the expansion taper, as discussed later
in this section.
5.16
Contraction Reach. Table 5.2 below is taken from Appendix B of the Hydraulic Reference
Manual. It summarizes the research published in Research Document 42 (USACE 1995)
with regard to the contraction rate (CR). It is similar to Table 5.1 but involves fewer factors. A
cell in the table is selected on the basis of longitudinal slope and roughness ratio. The values
in the cell are the appropriate range of CR values. The engineer selects a value within the
range and determines the length of the contraction reach. Every cell in the table includes a
CR value of 1. For this reason, and because in the research study the overall data set had
mean and median CR values both near 1, common practice is to routinely use a value of 1
for CR. This practice is usually acceptable, but for cases in which the bridge design is highly
sensitive to the amount of backwater (e.g., to comply with FEMA floodway regulations) it may
be advisable to use the table to select CR.
Table 5.2. Ranges of Contraction Rates, CR (after USACE 2010c).
Slope
nob/nc = 1
nob/nc = 2
nob/nc = 4
0.0002
1.4 3.6
1.3 3.0
1.2 2.1
0.001
1.0 2.5
0.8 2.0
0.8 2.0
0.002
1.0 2.2
0.8 2.0
0.8 2.0
Variable: S = slope, nc = channel Manning n, nob = overbank Manning n
Tidal Bridges. When bridges over tidal streams are analyzed with one-dimensional unsteady
flow models, cross section locations must accommodate flow in both directions. The
approach section during ebb tide conditions, in which flow is toward the ocean, will be the
exit section during flood tide conditions, in which the flow is away from the ocean. In this
case the CR and ER values should be the same, and should be a compromise between the
ER and CR values that would have been selected if the bridge were not tidal. In many cases
it is appropriate to use a value of 1.5 for both CR and ER.
5.5.3 Ineffective Flow Specifications at Bridges
Section 5.4.2 describes the Ineffective Flow feature in HEC-RAS. This feature is very useful
in modeling bridge crossings. Referring to Figure 5.8, the upstream and downstream
bounding sections are located beyond the side slopes of the embankment fills, and therefore
the cross section geometry reflects the floodplain, not the roadway. Ineffective flow areas on
the upstream and downstream bounding sections represent the presence of the highway
embankments. Figure 5.8 is an example of an upstream bounding section at a bridge. The
lateral position of the ineffective flow setting is set back from the abutment station by a
distance equal to the CR or ER value multiplied by the distance of the cross section
upstream or downstream from the bridge. The engineer assigns the elevation setting on the
ineffective flow setting based on the water surface elevation at which a significant amount of
discharge would flow over the top of the road embankment.
If intermediate cross sections are inserted in the transition reach between the bridge and the
approach section or between the bridge and the exit section, then those sections must
include ineffective flow specifications to reflect the taper of the contracting or expanding flow.
It is strongly recommended that the engineer plot the cross section lines and the taper lines
on a topographic map and/or aerial photograph in order to enter the ineffective flow
specifications correctly.
5.6 BRIDGE HYDRAULIC CONDITIONS
This section provides a qualitative description of the various types of flow conditions that can
exist at a bridge. Later sections explain the technical approaches to modeling the different
conditions.
5.17
5.18
Figure 5.10. Illustration of free-surface bridge flow Classes A, B, and C (USACE 2010c).
In some cases a weir does not accurately represent the flow over the roadway approaches.
This can occur either because the road is at or very near floodplain grade (in other words
there is little or no embankment fill) or because the downstream water surface elevation is so
high above the weir crest elevation that the weir control is drowned out.
5.6.3 Flow Submerging the Bridge Low Chord
A condition in which the water surface is above the highest point of the bridge low chord is
usually representative of orifice flow. When the low chord is submerged only at the upstream
edge of the superstructure, the orifice is considered free-flowing, and thus not affected by
tailwater. This condition is analyzed using the same approach as for an orifice (FHWA 1978)
and in this manual is referred to as "orifice bridge flow." Just as the headwater upstream of
an inlet-control culvert is not affected by conditions downstream of the culvert entrance, the
backwater upstream of a bridge operating under orifice bridge flow is not affected by
conditions downstream of the upstream edge of the superstructure.
5.19
Another type of orifice flow exists when the highest point of the low chord is submerged at
both the upstream and downstream edges of the superstructure. This type of flow is
analyzed using a formulation for a tailwater-controlled orifice (FHWA 1978). Just as the
headwater upstream of an outlet-control culvert is sensitive to conditions within and
downstream of the culvert barrel, the backwater upstream of a bridge operating under fullflowing or tailwater submerged orifice conditions is affected by conditions within and
downstream of the bridge waterway. For purposes of this manual, this condition is termed
"submerged-orifice bridge flow."
5.7 BRIDGE MODELING APPROACHES
This section explains the approaches and equations that are used to analyze the various
types of flow conditions that can exist at a bridge. The HEC-RAS Hydraulic Reference
Manual explains the approaches described in this section in greater detail (USACE 2010c).
The information presented in this section is predominantly taken from that source but with
much of the detail omitted. Except in the case of the WSPRO method, the discussion below
applies specifically to the region between the upstream bounding cross section and the
downstream bounding cross section. Upstream and downstream of this region, the energy
equation governs.
5.7.1 Modeling Approaches for Free-Surface Bridge Flow Conditions
The most commonly encountered free-surface bridge flow scenarios at bridges are Class A
conditions. The HEC-RAS program makes four modeling approaches available to users for
Class A flow: The Energy Method, the Momentum Balance Method, the Yarnell Equation
and the WSPRO Method. The four modeling approaches are described below. Figure 5.11
shows the cross section identifiers for reference to the bridge flow equations.
Energy Method. Chapter 3 describes the energy equation in detail. Section 5.3 describes its
general application to water surface profile calculations. When the energy equation is
applied to bridge hydraulics, the area occupied by the road embankments, abutments, bridge
deck and piers is subtracted from the effective flow area. The wetted perimeter is increased
to account for the sides of piers (often a minor effect) and the low chord of the bridge if it is in
contact with the flow. The low chord and pier sides can have a significant effect on the
wetted perimeter. Since the area is decreased and the wetted perimeter is increased, the
conveyance is often reduced significantly. The reduced conveyance, in turn, increases the
friction slope which increases the friction loss.
Momentum Balance Method. As discussed in Chapter 3, a momentum-based formulation
can be used to analyze open-channel hydraulics. The Momentum Balance Method is based
on the principle that the sum of forces acting in a given direction on a control volume is equal
to the mass of the water in the control volume multiplied by its acceleration. Hydraulics in the
bridge waterway can be solved using this force-balance approach in three steps. The first
step deals with the control volume between the downstream bounding section (designated
with subscript 2) and the downstream face of the bridge opening (subscript BD):
A BD Y BD +
2
BD Q BD
Q2
= A 2 Y 2 + 2 2 A P BD Y P BD + Ff ( 2BD ) W x ( 2BD )
gA BD
gA 2
5.20
(5.14)
BD BU
CR
ER
FLOW
Figure 5.11. Plan view layout showing cross section identifiers referenced by the bridge
hydraulics equations.
The second step operates on the control volume beneath the bridge superstructure:
A BU Y BU +
2
BUQBU
Q2
= A BD Y BD + BD BD + Ff (BDBU) Wx (BDBU)
gA BU
gA BD
(5.15)
The third step analyzes the force balance on the control volume between the upstream face
of the bridge opening (designated with subscript BU), and the upstream bounding section
(designated with subscript 3):
A3 Y3 +
A P BU Q 32
3 Q 32
Q2
1
= A BU Y BU + BU BU = A P BU Y P BU + CD
+ Ff (BU3 ) Wx (BU3 )
gA 3
gA BU
2
gA 3
(5.16)
=
=
Active flow area at the cross section denoted by the subscript, ft2 (m2)
Flow area obstructed by pier at the upstream and downstream faces of
the bridge opening, ft2 (m2) (see Figure 5.12)
5.21
Yi
Y P BU , Y P BD
Qi
i
=
=
Ff
Wx
CD
Vertical distance from the water surface to the centroid of the flow area
at the cross section denoted by the subscript, ft (m)
Vertical distance from water surface to the centroid of the pier area at
the upstream and downstream faces of the bridge opening, ft (m) (see
Figure 5.12)
Discharge at the cross section denoted by the subscript, ft3/s (m3/s)
Velocity weighting coefficient for momentum at the cross section
denoted by the subscript
External friction force acting on the control volume per unit weight of
water, ft3 (m3)
Component of the weight of water acting in the direction of flow, per
unit weight of water, ft3 (m3)
Drag coefficient for flow around the pier
YBU
YPBU
APBU
ABU
Figure 5.12. Cross section view illustrating the area and Y variables in momentum equation.
The user enters the drag coefficient, which is a function of the plan-view shape of the pier.
Table 3.3 provides guidance on the values of the pier drag coefficient. Because of the pier
drag coefficient, the Momentum Balance Method is sensitive to the hydraulic efficiency of the
pier shape. This is an advantage over the Energy Method, which does not provide a way of
accounting for streamlined pier shapes. The Momentum Balance Method is also the
preferred approach to computing the bridge hydraulics in Class B flow, because it is not
hindered by rapidly varied flow conditions.
5.22
Yarnell Equation. While the Energy Method and the Momentum Balance Method are
theoretically derived, the Yarnell Equation is strictly empirical. It is based on the results of
roughly 2600 flume experiments that were designed to test the relationship between the
change in water surface elevation caused by a pier and the size, shape, and configuration of
the pier in combination with varied flow rates. The resulting equation is:
2
V
H3 2 = 2K(K + 10 0.6)( + 15 4 ) 2
2g
(5.17)
where:
H3-2
=
=
V2
When using the Yarnell Equation, the engineer enters a pier shape coefficient. Table 5.3
provides appropriate values of the coefficient for various plan-view pier shapes. A
disadvantage of the Yarnell Equation is that, because it is strictly empirical, its application
should be limited to bridge sites that are similar in nature to the flume studies that were used
in the development of the equation. In practical terms, this means that the equation is only
appropriate for channels of generally regular cross section under approximately uniform flow
conditions, where piers are the only significant source of losses.
WSPRO Method. Beginning in the 1980s the FHWA developed and supported a watersurface-profile computer program, called WSPRO, that became the standard bridge
hydraulic analysis software for many state departments of transportation. The bridge
hydraulics approach from that program is now included as an available method in HEC-RAS.
The WSPRO Method is based on a standard-step solution of the energy equation, and is
similar to the Energy Method in most respects. Unlike the other three free-surface bridge flow
methods discussed here, the WSPRO Method works from the exit section to the approach
section, and not just between the upstream and downstream bounding sections. In general:
WS 4 +
4 V42
V2
= WS 1 + 1 1 + hL
2g
2g
(5.18)
where:
WS1, WS4
V1, V4
hL
Water surface elevation at the exit section (Cross Section 1) and at the
approach section (Cross Section 4), ft (m) (see Figure 5.14)
Velocity at the exit section and at the approach section, ft/s (m/s)
Sum of the energy losses between the exit section and the approach
section, ft (m) (see Figure 5.14)
5.23
Energy
Gradeline
2
V2
2g
H3-2
Water
Surface
Figure 5.13. Profile view with definitions of variables in the Yarnell Equation.
Table 5.3. Yarnell Pier Shape Coefficients (USACE 2010c).
Plan-View Shape of Pier
Drag Coefficient, CD
Semi-circular nose and tail
0.90
Twin cylinders with a connecting diaphragm
0.95
Twin cylinders without diaphragm
1.05
Triangular nose and tail with 90-degree angle
1.05
Square nose and tail
1.25
Trestle bent with ten piles
2.50
The WSPRO Method computes the energy losses incrementally in six parts: five increments
of friction loss and the expansion loss between the exit section and the downstream
bounding cross section (Cross Sections 1 and 2).
The five increments of friction loss cover the segments between Cross Sections 1 and 2;
between the downstream bounding section and the downstream face of the bridge opening
(Cross Sections 2 and BD); the segment under the bridge deck (Cross Sections BD and BU);
between the upstream face of the bridge opening and the upstream bounding section (Cross
Sections BU and 3); and between the upstream bounding section and the approach section
(Cross Sections 3 and 4). For each of the first four segments, the friction loss is calculated
using the following general equation:
5.24
hfseg =
L segQ2
K dK u
(5.19)
where:
h fseg
Lseg
Q
Kd, Ku
=
=
While not readily recognizable as such, the Q2/KdKu portion of Equation 5.19 is the geometric
mean friction slope between the two cross sections (see Equation 5.6).
The friction loss for the upstream-most segment is slightly different:
hf3 4 =
L av Q2
K 3K 4
(5.20)
where:
hf3 4
K3, K4
Lav
=
=
Energy
Gradeline
hL
WS1
hf3 4
he
4 V4
2g
1V1
2g
Water
Surface
Figure 5.14. Profile view with definition of terms in the WSPRO Method.
5.25
WS4
HEC-RAS computes the effective average flow length in Equation 5.20 as the average length
of 20 equal-conveyance stream tubes that flow from Cross Section 4 to Cross Section 3 on
theoretical curvilinear paths. The details of the assumed stream tube flow paths are
explained in Appendix D of the HEC-RAS Hydraulic Reference Manual and in the WSPRO
User's Manual (FHWA 1986).
By default, the WSPRO Method does not include the standard contraction and expansion
losses from the energy equation (e.g. the contraction or expansion coefficient multiplied by
the absolute difference in velocity head between two cross sections) in the energy losses
from the exit section to the approach section. Expansion losses between Cross Sections 1
and 2 are accounted for using the following equation:
he =
2
A1
A1
Q2
2
2
+
1
1
2
2
2gA 12
A 2
A 2
(5.21)
where:
he
A1, A2
1
1
2, 2
=
=
=
=
=
Appendix D of the HEC-RAS Hydraulic Reference Manual (USACE 2010c) explains in detail
the empirical discharge coefficient C mentioned in the definition of the variables 2 and 2.
5.7.2 Selection of Free-Surface Bridge Flow Modeling Approach
The Energy, Momentum Balance and WSPRO Methods are all suitable to a wide range of
conditions. Among these three, the Momentum Balance Method is unique in accounting for
the pier drag as a function of pier shape. Therefore the Momentum Balance Method is
recommended in cases where piers are the predominant energy loss factor and especially
when the pier geometry is somewhat streamlined.
The Energy and WSPRO Methods are both effective in conditions where friction loss and the
effects of constriction are predominant. In most cases the results of the two methods, when
applied correctly under the same conditions, are very similar in terms of the energy grade
line and water surface elevation upstream of the bridge. Only the WSPRO Method, however,
accounts for different types of abutments geometries (for instance spill-through abutments
vs. vertical abutments with wing walls). The Momentum Method also typically performs well
in situations where the constriction is the predominant loss factor, and has the advantage
that it can better accommodate rapidly-varied flow, which is important in Class B and Class C
free-surface bridge flow.
Because of its empirical derivation, the Yarnell Equation is suitable only for Class A cases in
which the geometry of the waterway is generally uniform and regular, and without a great
degree of constriction. It can be expected to perform well in analyzing bridges over manmade channels such as irrigation canals or engineered flood control channels.
5.26
Ideally engineers modeling Class B and Class C flow conditions should employ the
Momentum Method for the reasons mentioned earlier. The Energy Method is also an
acceptable approach, although less ideal.
5.7.3 Modeling Approaches for Overtopping and Orifice Bridge Flow
The HEC-RAS program makes two different approaches available to the user for modeling
overtopping and orifice bridge flow conditions: Energy Method and Pressure and Weir
Method.
Note that in HEC-RAS terminology, orifice and submerged-orifice bridge flow
conditions are termed "pressure flow."
Energy Method. Just as described above for free-surface bridge flow modeling, the Energy
Method simply continues the standard-step solution of the energy equation through the
bridge structure and vicinity. It accounts for the blockage caused by the road embankments,
abutments, bridge deck and piers simply by reducing the conveyance. If the water surface is
high enough to overtop the road, the program will treat the flow area above the road as
conveyance area, but not as a weir. When the Energy Method is used, the quantity of
overtopping flow will not be computed or reported. If the low chord is submerged, the added
wetted perimeter will have a negative effect on conveyance, but the program will not attempt
to compute orifice conditions.
Pressure and Weir Method. If the user has specified the Pressure and Weir Method, then
the broad-crested weir equation is be used to compute the quantity of any overtopping flow.
One of two orifice equations is used when orifice or submerged-orifice bridge flow is
detected.
Overtopping (Weir) Flow. The technique for computing weir hydraulics in the case of flow
over the road or bridge deck is very similar to the approach that was described and
recommended in HDS 1 (FHWA 1978). Figure 5.15 depicts the condition of flow overtopping
a roadway embankment. The broad-crested weir equation is:
Q W = CLH3 / 2
(5.22)
where:
Qw
C
L
H
=
=
=
=
The discharge coefficient for a broad-crested weir generally ranges between 2.6 and 3.1. A
bridge deck is not an ideal broad-crested weir and it is generally recommended that the lower
value of 2.6 be used for the discharge coefficient where increased resistance to flow caused
by obstructions such as bridge railings, curbs, and debris are present. HDS 1 provides a
curve of the C value versus head on the roadway. That curve is reproduced here as Figure
5.16.
5.27
Q=CLH3/2
Water Surface
H
Road Embankment
Q = CD A BU
where:
Q
CD
ABU
Z V2
2g Y3 + 3 3
2
2g
=
=
=
Y3
V3
3
=
=
=
1/ 2
(5.23)
Discharge under bridge deck, through the bridge waterway, ft3/s (m3/s)
Orifice flow discharge coefficient
Net flow area at the upstream face of the bridge opening, under the low
chord, ft2 (m2)
Height of the bridge opening from the highest point on the upstream low
chord to the mean riverbed elevation, ft (m)
Hydraulic depth at the upstream bounding section (Cross Section 3), ft (m)
Velocity at the upstream bounding section, ft/s (m/s)
Kinetic energy distribution coefficient at the upstream bounding section
5.28
Figure 5.16. Guidance on discharge coefficients for flow over roadway embankments,
from HDS 1 (FHWA 1978).
Y3
5.29
The head forcing the flow through the orifice is defined as the vertical distance from the
energy grade line upstream of the bridge to roughly the vertical center of the bridge opening
height (the elevation halfway up the Z dimension). The value of the discharge coefficient CD
is related to the ratio of low chord submergence (Y3/Z) as shown in Figure 5.18.
Figure 5.18. Relationship between orifice bridge flow discharge coefficient and
submergence of the low chord, from HDS 1 (FHWA 1978).
Note that the curve is shown as a dashed line to the left of Y3/Z=1.1. This region of the curve
represents a transition zone in which orifice flow conditions have not been reliably
established. At this submergence level, the flow could be expected to vary between openchannel and pressure flow conditions, and the orifice equation might not be reliable.
Figure 5.19 illustrates the case of submerged-orifice bridge flow. The equation for this case
is:
Q = CA 2gH
(5.24)
where:
Q
C
A
H
=
=
=
=
Discharge under bridge deck, through the bridge waterway, ft3/s (m3/s)
Discharge coefficient for submerged-orifice bridge flow (usually 0.8)
Net flow area of the bridge opening, ft2 (m2)
Vertical distance between the upstream energy grade line and the
downstream water surface, ft (m)
In the submerged-orifice case, the head is measured from the upstream energy grade line to
the downstream water surface elevation, reflecting that the downstream conditions have a
direct effect on the backwater. Field data reported in HDS 1 indicated that the values of C for
submerged-orifice bridge flow range from 0.7 to 0.9. Common practice, encouraged by HDS
1, is to use a value of 0.8 for C.
5.30
Water Surface
H
Y3
Y2
5.31
Similar to the previous case, if overtopping is occurring and the water surface difference
is substantial across the road alignment, then true weir flow conditions are likely. Again
the Energy Method overestimates the backwater caused by such cases.
Engineers occasionally encounter situations that are borderline cases, where the decision
between the two methods is not clear cut. One such example is a case in which the bridge
low chord is submerged with a low degree of submergence (Y3/Z < 1.1) and there is no
overtopping flow. The flow conditions are not firmly in the realm of orifice flow at this degree
of submergence and the Energy Method may be more appropriate. In this case it is
recommended to conduct the analysis with both the Pressure and Weir and the Energy
Methods. Use the more conservative of the two if the two results do not differ greatly. Use
the Energy Method result if the two results differ by a significant amount.
The recommendations above call for making decisions based on the observed flow
conditions. Usually the engineer will not be able to anticipate at the outset what types of flow
conditions will be observed in the model results. The modeling process, therefore, requires
some iteration by the engineer before arriving at a final analysis model. The recommended
practice is to make an initial model run with the Pressure and Weir Method selected. If the
model results show that overtopping is occurring, the engineer should identify whether the
weir crest is submerged, and if so, by how much, and then decide whether the overtopping
flow is truly functioning as weir flow. If so, then the Pressure and Weir Method is appropriate.
If not, then the Energy Method should be used. If no overtopping is occurring but there is
orifice or submerged-orifice bridge flow, then the engineer should decide between the
Pressure and Weir and Energy Methods based on the degree of low chord submergence.
5.32
(5.25)
where:
X
=
=
(5.26)
5.33
FLOW
a proj b proj
=
=
=
A skew angle of 30 is identified by HDS 1 as a practical maximum for analysis by the bridge
opening adjustment concept described here. The engineer should consider a different
modeling approach, such as two-dimensional analysis, when the skew angle exceeds 30.
Section 5.9 discusses some important limitations related to the one-dimensional treatment of
skewed crossings.
5.8.2 Crossings with Parallel Bridges
Parallel bridges are a common occurrence when streams are crossed by divided highways.
Figure 5.21 shows an example of a parallel bridge crossing. Hydraulically the two bridges are
in series. Physical modeling results reported in HDS 1 show that two identical bridges in
series and in close proximity to each other produce about 1.3 to 1.5 times the backwater
caused by one bridge alone, depending upon the distance between the two bridges (see
Figure 5.22). The maximum clear distance between the bridges in the study cited was 9
bridge deck widths (Ld/l equal to 11 in Figure 5.22). One likely reason for the total backwater
being less than twice the single-bridge backwater is that the full contraction and re-expansion
of the flow would have occurred only once (contracting upstream of the upstream bridge and
re-expanding downstream of the downstream bridge).
5.34
Figure 5.22. Backwater multiplication factor for parallel bridges (from FHWA 1978).
5.35
Except in very rare cases it is acceptable to model the two bridges as separate structures in
series, but this approach does require additional effort by the engineer that may not be
necessary. Depending upon the purpose of the analysis, it may be acceptable to model two
parallel, identical bridges as a single bridge. If the two bridge decks are within just a short
distance of each other, then treating them as a single bridge with the deck width equal to the
sum of the two decks is appropriate. If the two bridges are farther apart, it might be more
appropriate to enter the deck width as the total distance from the upstream edge of the
upstream bridge to the downstream edge of the downstream bridge, so as not to completely
neglect the losses that will be generated in the gap between the two structures.
Many scenarios make it advisable or necessary to model the two bridges as separate
structures in series, each with its own bounding cross sections and bridge data
specifications. Examples of such cases are:
When the purpose of the model is to develop the hydraulic design of the two structures
and the engineer needs refined hydraulic information to apply to each structure
independently for freeboard determination, scour evaluation, etc.
When the flow can re-expand and consequently re-contract between the two bridges, as
could occur if there is substantial distance between the bridges and the ground between
the divided roadways is low enough to allow it
When submerged-orifice bridge flow is a possibility, since the backwater of the upstream
bridge is very sensitive to conditions downstream in a submerged pressure flow condition
When the two bridge structures are not identical in terms of span lengths, pier geometry,
deck profile, etc.
3R
4R
5R
6R
SPLIT RIGHT
FLOW
SPLIT LEFT
3L
4L
5L
6L
5.36
To model a split flow condition, the engineer defines a separate model reach for each flow
path. Each reach has its own series of cross sections and its own flow rate. The flow rate for
each reach is defined at the point where the flow paths diverge (usually defined by a
branching junction in HEC-RAS). The correct apportionment of the total flow between the
separate reaches is not known at the beginning of the analysis, but is determined through a
trial-and-error process. The process of finding the correct flow apportionment is based on the
principal that all separate reaches branching from a single point are expected to yield the
same energy grade line elevation at the point of divergence. The analysis progresses as
follows:
1. A trial flow rate is assigned to each reach, with the constraint that the sum of the
individual reach flow rates must equal the total flow upstream of the point of divergence.
2. Using the assigned flow rates, a water surface profile is computed for each separate
reach.
3. The resulting energy grade line elevations at the upstream ends of all of the separate
reaches are compared.
4. If all of the resulting energy grade line elevations match each other within the desired
tolerance, which is usually 0.05 feet (0.015 m) or less, then the correct flow
apportionment has been found and the analysis proceeds beyond the split reaches.
5. If the disagreement between the resulting energy grade line elevations exceeds the
desired tolerance, then the flow rates are reduced in the reaches that produced the
highest energy grade lines and increased in those that yielded the lowest values, and the
process begins again at step 2.
This process can be used whether or not the separate reaches recombine downstream. If
they do recombine, then the total energy loss in all reaches must be the same from the
branching junction to the confluence junction. If the reaches do not recombine, then the total
energy loss in all reaches might not be the same, but in the final solution they must all
produce the same energy grade line at the point of divergence.
The HEC-RAS program facilitates the modeling of split flow reaches. The Junction
Optimization feature can be activated by the engineer at any branching flow junction. If the
feature is activated the program automatically performs iterations and checks for
convergence of the energy grade lines until the correct flow apportionment is found. The
same principles and approaches used in solving the flow apportionment in split reaches are
used in the analysis of multiple-opening crossings.
5.8.4 Crossings with Multiple Openings in the Embankment
Some crossings require relief bridge openings or culverts through the embankment in
addition to the main bridge opening. Particularly wide floodplains and those with separate
side channels are examples of sites that might require multiple openings. Figure 5.24
illustrates a multiple-opening crossing. Similar to split reaches, the multiple-opening scenario
presents a special challenge in one-dimensional analysis. The analysis must correctly
determine the apportionment of the flow to each opening in the embankment. The engineer
is encouraged to consider the use of two-dimensional analysis for multiple-opening
situations.
5.37
FLOW
STAGNATION
POINT
STAGNATION
POINT
RS=5.4
#1
#2
225
Legend
Ground
220
Ineff
Elevation (ft)
Bank Sta
Stag Limit
215
210
205
200
500
1000
RS=5.4
#1
2000
1500
2000
#3
#2
225
1500
Elevation (ft)
220
215
210
205
200
500
1000
Station (ft)
5.39
LATERAL WEIR
FLOW OVER ROAD
3
7
8
FLOW
Q x1 x 2 =
2C
[(a 1 x 2 + C1 ) 5 / 2 (a 1 x 1 + C1 ) 5 / 2 ]
5a 1
(5.27)
where:
Q x1 x 2
The weir overflow along the lateral weir segment, ft3/s (m3/s)
C
a1
=
=
x1
x2
C1
=
=
=
5.40
The standard weir coefficients for a weir crest perpendicular to the floodplain flow are often
inappropriate for lateral weir flow calculations because the momentum of the floodplain flow
causes the overtopping flow to cross the weir crest at an oblique angle. HEC-RAS offers the
option to use Hager's (1987) equation to compute the weir coefficient:
C=
1 W
3
C0 g
5
3 2y W
0 .5
0 .5
3(1 y )
1
(
S
)
y W
(5.28)
where:
C0
W
=
=
S0
5.41
Water Surface
A
Energy
Ground
Flow
Figure 5.27. One-dimensional model cross sections.
5.9.2 Total Energy and Flow Distribution
Energy varies throughout the cross section because velocity is not constant throughout the
cross section. Cross section A-A in Figure 5.27 shows local energy (WS + V2/2g) computed
from local water surface and velocity. Total energy is the local energy integrated for the
entire cross section. Therefore, another assumption is that the total kinetic energy computed
5.42
V 2
is representative for the cross section. This implies that flow is distributed within
2g
the cross section proportional to conveyance. It also implies that every point within the cross
section has the same energy slope. These are the assumptions that allow the program to
estimate velocity and flow distribution between floodplain and channel, or between different
locations within the floodplain or channel. The assumption is accurate for uniform flow and
normal depth, but in locations of significant flow curvature the assumption breaks down.
from
5.43
5.44
CHAPTER 6
TWO-DIMENSIONAL BRIDGE HYDRAULIC ANALYSIS
6.1 INTRODUCTION
Chapter 4 describes the differences between one-dimensional and two-dimensional hydraulic
analyses. Most bridge hydraulic studies have used one-dimensional analysis methods that
are described in detail in Chapter 5, though two-dimensional models are being used
frequently, especially for complex situations. As the use of two-dimensional models
becomes more commonplace, they will, inevitably, be used for all but the most
straightforward bridge hydraulic conditions. This chapter provides information and guidance
on the use of two-dimensional models for bridge hydraulic analysis.
In one-dimensional modeling, the standard step solution of the energy equation is most
frequently used for hydraulic analysis. For two-dimensional modeling, the momentum
equation (Newton's second law of motion, F = ma) is applied to a control volume in
conjunction with the continuity equation. Figure 6.1 illustrates a control volume of flowing
water in three-dimensional space and includes the primary forces acting on the control
volume in two-dimensions. The calculated variables of velocity and depth are also shown.
In two-dimensional models, vertical velocity components are considered as negligible and
hydrostatic pressure is assumed. Velocity is a vector quantity that can be expressed as a
magnitude and direction or as the x and y velocity components U (x direction) and V (y
direction). The elevation of the bed (Zb) and water depth (H) vary over the area. The force
variables shown in Figure 6.1 are pressure (P) at the control volume horizontal surfaces,
water weight (W), bed shear stress components (b), and water surface shear stress (s). For
a set of unbalanced forces, the mass associated with the control volume will accelerate. As
will be discussed in the next section, these variables are the primary set of forces acting on
the control volume, though others are included depending on the problem.
6.1
(6.1)
where:
Zw
qx
qy
qm
=
=
=
=
=0
y
x
qy
t
2
Z
qy g 2 qy qx
H Pa
+ gH b +
+ H +
+ qx
y H 2 x
y
H
y
(H yy ) (H yx )
1
+ by sy
=0
y
x
where:
H
Zb
g
Pa
bx, by
sx, sy
xx, xy
yy, yx
=
=
=
=
=
=
=
=
=
=
=
6.2
(6.2)
(6.3)
Acceleration Terms
Coriolis term
Shear
Stresses
Caused by
Turbulence
Atmospheric
Pressure gradient
Water Weight
(gravity & bed slope)
Hydrostatic
Pressure
Z
P
(H xx ) (H xy )
q2x qx qy
H
+ H
+
+ H b + H a + bx sx
qy = 0
H
x H y
x
x
x
x
y
Force Terms
F x
Pseudo
Force Term
qx
Convective
Acceleration
Mass
Local Acceleration
The terms of Equations 6.1 and 6.2 are mass times acceleration or force terms in Newton's
second law of motion, F = ma, as shown in Figure 6.2. Figure 6.2 is the x direction
(Equation 6.2) and there are equivalent terms in the y direction equation. The equation
shown in Figure 6.2 has been rearranged and multiplied by mass () to clearly indicate mass
times acceleration terms versus force terms. The acceleration terms are local acceleration
(time) and convective acceleration (converging or diverging stream lines). The force terms
include changing hydrostatic pressure due to changing flow depth, the component of water
weight acting on the sloping bed, atmospheric pressure gradient, bed shear stress, water
surface shear stress due to wind, shear stresses caused by turbulence, and the pseudo force
term due to the Coriolis Effect. The bed shear stress is evaluated using the Manning or
Chezy relationships, though Manning is more commonly used. Surface shear is related to
wind speed. Turbulence shear relates to turbulence exchange and horizontal diffusion of
momentum. The Coriolis Effect is due to the fact that the model represents an area on the
rotating earth. There is an apparent force which is caused by the resulting angular
acceleration. Some of these forces are negligible in many river applications, including
Coriolis, surface shear, and atmospheric pressure gradient, which apply most often to large
bodies of water in tidal applications. The solution techniques for Equations 6.1 through 6.3
include the application of the finite difference method and finite element method.
=0
(from F = ma)
6.3
mid-side nodes out of alignment with the corner nodes. The FST2DH model (FHWA 2003)
also includes a quadrilateral element shape with a center node in addition to the mid-side
and corner nodes. Figure 6.3 illustrates a variety of element types and shapes. The element
sides do not need to align with the x- or y- directions, they do not need to have consistent
size or orientation, and a mixture of triangular and quadrilateral elements are allowed. The
unstructured mesh forms the geometric framework for the hydraulic computations.
Corner node
Mid-side node
Center node
Figure 6.3. Element types and shapes.
Figure 6.4 is an example of a mesh layout using triangular and quadrilateral elements. The
elements are arranged based on several criteria, which include:
Topography and bathymetry are represented by assigning elevations to the nodes. Land use
is represented by varying roughness conditions (Manning n) by assigning material types to
the elements. Figure 6.4 illustrates how triangular elements can be used to transition from
large to small elements and to represent curved features. This allows for areas with greater
topographic, land use and velocity variability to have greater detail. The velocity vector (xand y- components) and flow depth are computed at each corner, mid-side, and center node.
6.4
Crop
Dense
Trees
Cleared
Channel
Cleared
Dense
Trees
Light
Trees
6.5
1-D Connection
Road Alignment
Cell Disabled
6.6
6.7
996
6.8
6.9
>130
<10
<10
>130
<10
5.0
10.0
10.0
5.0
5.0
10.0
5.0
10.0
10.0
5.0
6.10
Disabled Elements
Void in Mesh
Pier
Pier
Nodes
Figure 6.11. Pier drag force.
6.12
Increased Flow Resistance Method. Another approach to including the pier drag force is to
increase the Manning n of the element. The force caused by pier drag is equated to an
increase of shear stress so the total force is equivalent. The hydraulic engineer needs to
compute the area of piers in an element or group of elements and apply the following
equation to estimate the effective Manning n (ne) that accounts for the additional force. One
disadvantage of this approach is that angle of attack is not directly accounted for in the
model but needs to be included when determining the pier width. The advantage of this
approach is that the drag from a large number of piers can be incorporated into the model by
adjusting the Manning n of some or all of the bridge elements. As with the additional force
method, an obstruction is not modeled and flow redistribution may not occur. The effective
Manning n is computed from:
ne = n +
2
C A
2g A
K u2 y 1 / 3
(6.4)
where:
ne
n
Cd
Ap
AE
y
g
Ku
=
=
=
=
=
=
=
=
6.13
Weir Elevation
Weir Connection
Culvert Connection
6.14
Velocity,
ft/s (m/s)
9 (2.7)
8 (2.4)
7 (2.1)
6 (1.8)
5 (1.5)
4 (1.2)
3 (0.9)
2 (0.6)
1 (0.3)
0 (0.0)
6.15
difficult for hydraulic engineers to assess whether an upstream bridge or pier creates
adverse conditions for a downstream structure. Therefore, when structural elements are not
well aligned, or if questions exist regarding pier placement at an adjacent structure, twodimensional models should be used. Depending on flood elevations, embankment heights
and the amount of backwater created by the crossing, road overtopping may occur for one
embankment and not another. These complex hydraulic conditions can be analyzed directly
in two-dimensional models.
As with modeling multiple openings along an embankment, there are no additional
requirements for modeling multiple openings in series in a two-dimensional model than those
required for modeling single openings.
Geometry, land use, roughness, boundary
conditions, and model limits must be accurately represented. As compared with onedimensional modeling, there are no requirements for estimating flow expansion and
contraction between bridges or assigning ineffective flow areas, so the need for judgment is
reduced. Therefore, a more accurate distribution of flow within each of the bridge openings
is computed.
6.8 UPSTREAM FLOW DISTRIBUTION
As illustrated in the model in Figure 6.7, the results of which are shown in Figure 4.3, flow
distribution may be affected by upstream structures. These structures may be river control
structures, countermeasures, or other bridge openings as discussed in the previous section.
In one-dimensional subcritical models all computations progress from downstream and flow
is distributed based on conveyance. Therefore, the effects of upstream controls on flow
distribution cannot be simulated in one-dimensional models other than by manipulating
conveyance through Manning n or assigning areas as ineffective. Although the downstream
water surface boundary condition is still required as a control for subcritical two-dimensional
models, upstream impacts on flow distribution are well-simulated in two-dimensional models.
6.9 SPECIAL CASES IN TWO-DIMENSIONAL MODELING
Chapter 5 includes special cases of one-dimensional modeling that fall outside the typical
model application. These include skewed crossings, parallel crossings, multiple openings
and other less common applications. Most of these situations are not considered as special
applications in two-dimensional models because the assumptions required by onedimensional models are not required in two-dimensional models. This section provides
guidance on the use of two-dimensional models for some of these cases, and compares and
contrasts the use of one- and two-dimensional models for them.
6.9.1 Split Flow
Figure 6.14 is a two-dimensional model representation of the one-dimensional split flow
model shown in Figure 5.23. In a one-dimensional model, separate reaches are required for
the two split flow reaches and two more reaches are required upstream and downstream of
the split flow reaches. In the one-dimensional representation, a trial and error process is
used to apportion flow between the two reaches until an energy balance at the upstream
combined-flow cross section is achieved. Two-dimensional modeling provides a better
depiction of split flow hydraulics because the assumption of energy balance at a particular
location is not made. In essence, the two-dimensional model is always using an iterative
process to apportion flow throughout the network until the equations of motion are satisfied.
Therefore, solution to the split flow problem is intrinsic to two-dimensional hydraulic analysis.
The assignment of a specific location for energy balance is not a requirement in two6.16
dimensional modeling, nor is it even possible to make that assignment. Another advantage
of two-dimensional modeling is that if flow over the island from one "reach" to the other
occurs, the two-dimensional model would not need a special lateral connection that is
required by the one-dimensional model.
Materials
Channel
Floodplain
Island
Flow
6.17
6.9.4 Debris
There is no automated method for including debris on piers in an FST2DH model as there is
in HEC-RAS. However, the area of the debris blockage can be included with the pier
dimensions and an increased force will be computed using the additional drag force method
or increased flow resistance method.
6.10 TWO-DIMENSIONAL MODELING ASSUMPTIONS AND REQUIREMENTS
6.10.1 Gradually Varied Flow
Although two-dimensional modeling provides a much more complete analysis of bridge
hydraulics than one-dimensional modeling, especially as it relates to flow distributions and
lateral velocity components, two-dimensional models do not account for vertical velocities
and accelerations. Therefore, the flow is assumed to have a hydrostatic pressure
distribution, vertical velocities are neglected, and flow circulation is not simulated. If these
flow features are an important aspect of the flow hydraulics, such as detailed analysis of flow
at a pier, then three-dimensional models, CFD models, or physical models are required.
6.10.2 Flow Distribution and Water Surface at Boundaries
Two-dimensional models make various assumptions about flow distribution and water
surface elevation at boundaries. Water surface boundaries are usually treated as a level
water surface. This may not be an accurate representation, so, as with one-dimensional
models, it is best to have the downstream boundary located well away from the point of
interest. Although it is possible to enter a varying water surface boundary, the data
necessary to establish the input is often not available. The FST2DH model will usually
distribute water at the upstream boundary very reasonably. Other models may or may not do
as well. In any case, the model results should be evaluated to make sure that the upstream
boundary is not unduly influencing the solution. The upstream boundary should also be
located well away from the location of interest. As a rule-of-thumb, the upstream and
downstream boundaries should be at least one floodplain width upstream and downstream of
the bridge crossing. If flow is not fully expanded at the boundaries and largely onedimensional, then the model extent should be increased.
6.10.3 Model Step-Down and Convergence
Unlike one-dimensional models, which have a control either at the downstream or upstream
boundary depending on flow regime, two-dimensional models do not compute flow by
progressing from one boundary to another. The starting condition for most two-dimensional
models is a uniform pool of water that inundates the entire model domain. Once a solution is
achieved for this condition, the model head boundary is stepped down by small increments
until the desired water surface boundary is achieved. The model must achieve a reasonably
stable and converged solution for each intermediate run. This process can be tedious and at
times difficult if the model becomes unstable. There are many approaches to achieving an
efficient step-down process and a stable, yet accurate target condition. These include
stepping down water surface elevation at a low discharge and then stepping up discharge,
and using high Manning n and viscosity terms during the step-down process and then
stepping down these coefficients. The SMS software has automated run-control that
includes various step-down procedures (Aguaveo 2011).
6.18
Once the model water surface and discharge boundary conditions, and other input variables
including Manning n and viscosity terms, have reached the target values, the model must be
run for a sufficient number of iterations to converge on a numerically valid solution.
Convergence criteria related to water surface, velocity, or unit discharge can be set to halt
program execution once the criteria are met.
6.10.4 Wetting and Drying
Most two-dimensional models require all the nodes of an element to be wetted for that
element to be included in the flow computations. If a single node is dry, then the entire
element is removed from the computational network. Dry elements along a discharge or
water surface boundary should be avoided as it may cause model termination. The wetting
and drying process can create numerical instabilities but also may exclude areas with
significant conveyance when the elements are large. It is desirable to have a relatively
smooth boundary along the wet/dry boundary.
There are several techniques for maintaining model stability when areas of the network are
dry. These include incorporating additional network refinement and using depth-variable
Manning n and increasing its value for shallow depths. Some models, including FST2DH
and RMA2, allow the engineer to assign porosity to the ground. By assigning a low porosity
value, this approach allows for a very small amount of flow even at nodes where the water
surface is below the node elevation.
6.19
6.20
CHAPTER 7
UNSTEADY FLOW ANALYSIS
7.1 INTRODUCTION
Almost all flow in rivers and streams is to some extent unsteady, i.e., it changes with time.
Also, the rate of flow and the depth usually vary along the river. In many applications, flow
may be assumed to be uniform along a short reach of the channel. Among the most
important causes of unsteady flow are the following:
1. Runoff from precipitation (rainfall event and/or snowmelt); when depth and velocity of flow
in a river change rapidly with time
2. Unsteady or transient flows released from reservoirs during operations for flood control,
hydropower generation, recreation, and wildlife management, etc.
3. Tidal-generated waves (astronomical tides)
4. Dam-break floods
5. Wind-generated storm surges or seiches
6. Landslide-generated waves
7. Earthquake-generated tsunami waves
8. Irrigation flows affected by gates, pumps, diversions, etc.
There are several computer models that have been developed for simulating onedimensional flow. Fread of the National Oceanic Atmospheric Administration's National
Weather Service (NOAA's NWS) developed two unsteady flow models having the capability
to simulate flows through a single stream or a system of interconnected waterways.
DWOPPER was the original model and was later replaced by FLDWAV (Jin and Fread
1997). The HEC-RAS model (USACE 2010c) incorporates the UNET (USACE 2001)
unsteady flow algorithms for a full network of natural and constructed channels. The HECRAS model is the most widely used 1-D model in the US.
Figure 7.1 illustrates the basic properties of flow hydrographs. The downstream flow
hydrograph exhibits a delay (travel time) and is reduced (attenuation) when there is no
additional flow contributions between the upstream and downstream locations. Hydrologic
routing focuses on the discharge hydrograph. For bridge hydraulics, hydrodynamic
simulation is preferred because hydraulic variables (velocity, water surface elevation, depth,
etc.) are computed throughout the channel reaches represented in the model domain.
7.1.1 Unsteady Flow Equations Saint-Venant Equations
The discussion in this section emphasizes the one-dimensional equations and applications;
however the two-dimensional equations also will be presented. Solutions to one-dimensional
flow problems are conveniently viewed in a three-dimensional coordinate space, in which two
of the axes are distance along the channel and time, x and t, respectively, as shown in
Figure 7.2. The third coordinate axis corresponds to the solution, such as discharge Q(x,t).
Similar three-dimensional surfaces can be used to represent the variation in depth Y(x,t),
water surface elevation h(x,t), or velocity V(x,t). From Figure 7.2, the initial hydrograph is
given through time at the upstream cross section at x = x0. As the flood wave travels
downstream, the hydrograph is attenuated and lagged in time. Figure 7.2 also illustrates that
the condition at a location x >0 will not change for some amount of time (lag time) before the
initial upstream change in flow propagates downstream.
7.1
Travel Time
Attenuation
Inflow
Hydrograph
Discharge
Routed Downstream
Hydrograph
Time
t
tn
Q (t=0)
Lag Time
Figure 7.2. Unsteady solution of discharge versus x and t.
7.2
The initial conditions for an unsteady flow model are first a solution of the steady state flow,
Q at time t = t0 (i.e., steady state water surface profile computation), and the inflow
hydrograph at the most upstream channel cross section. In Figure 7.2, this indicates that
flow and depth would be known along the x axis for time t = t0 and the flow or depth would be
known at the upstream cross section x = x0.
Looking at only the x-t plane in Figure 7.3, known water surface elevations and flow along
the x-axis are indicated by the squares at time t = t0 and the known inflow hydrograph
ordinates at cross section x0 are illustrated by the circles. The unknowns are discharge and
depth along the channel at successive time lines until the entire surface is known. In Figure
7.2 the solution has reached time tn.
x
Figure 7.3. x-t plane illustrating locations for known and computed values.
The equations governing the flow of water in general are based on known physical principles,
conservation of mass and momentum. In unsteady flow analysis the continuity and the
momentum equations must be solved explicitly because the flow and the elevation of the
water surface are both unknown.
7.1.2 Unsteady Continuity Equation
The continuity equation for one-dimensional unsteady flow considers water that accumulates
or depletes in a control volume (storage). In steady flow the conservation of mass can be
written as Q = AV where the discharge is constant and the only unknown is the water
surface. Consider a short length (x) of channel as shown in Figure 7.4. The derivations
make the following assumptions (Chaudhry 2008):
1. The pressure distribution is hydrostatic
2. The channel bottom slope is small so that the flow depth measured vertically is almost
the same as the flow depth normal to the channel bottom (i.e., sin tan , where
is the angle between the channel bottom and the horizontal datum)
3. The velocity distribution at the channel cross section is uniform
4. The channel is prismatic; that is, the channel shape remains unchanged with distance
7.3
5. The friction losses in unsteady flow may be computed using the empirical formulas (i.e.,
the Manning equation) for steady-state flows
1
2
Q1
H
Q2
Datum
x
Figure 7.4. Definition sketch for continuity equation.
Assuming that the lateral inflow into or out of the reach is represented by qx, the continuity
equation can be written as:
Q 2 Q1 =
Q
x
x
(7.1)
The partial derivative is necessary since Q is changing with both time and distance along the
channel. Given that h represents the water surface elevation above the datum (h = z + y),
the volume of water between sections 1 and 2 is increasing at the rate T(h/t)t. From the
conservation of mass, the change in the flow must be equal to the change in the channel
storage, and for a short length of channel x where T is the top width of the water surface as
shown in Figure 7.5, (A/t)x T(y/t)x.
T = Topwidth
dy
dA
dA = Tdy
7.4
(7.2a)
V
y
y
+ VT
+T
=q
x
x
t
(7.2b)
In order to account for off-channel storage, Fread (1981) wrote the continuity equation as:
Q ( A + A 0 )
+
=q
x
t
(7.2c)
where A is the active cross-sectional area of flow and A0 is the inactive (off-channel storage)
cross-sectional area.
7.1.3 Dynamic Momentum Equation
Applying Newton's second law to the elemental length between Sections 1 and 2 of Figure
7.4 yields:
dV
V V
+
= ma = Ax
= Ax V
t
dt
(7.3)
The net forces causing flow down slope in Figure 7.4 are (1) the force resisting the shear
force (i.e., action versus reaction) and (2) the difference in the hydrostatic forces acting on
the element. Other forces may be present for special cases. These could include pier drag
forces and wind surface shear. The shear force can be represented by 0WPx and the
difference in the hydrostatic forces in the downstream direction is given by Ah assuming
the water surface slope and channel bed slope are small.
Substituting these forces into Equation 7.3 gives the equation of motion:
V V
Ah 0 WP x = Ax V
+
t
x
(7.4)
0 = R
+
+
x g x g t
(7.5)
Sf =
z y V V 1 V
x x g x g t
(7.6)
7.5
S f = S0
y V V 1 V
x g x g t
(7.7)
Solved together with boundary conditions, Equations 7.2 and 7.7 are the complete dynamic
wave equations for one-dimensional unsteady flow. Chapter 6 provides the complete set of
dynamic equations for two-dimensional flow analysis and includes additional force terms that
are difficult to represent in one-dimensional models.
Dynamic Wave Equation Terms. Meanings of the various terms in the dynamic wave
equations are as follows (Henderson 1966):
Continuity equation:
Momentum equation:
Depending upon the relative importance of the various terms of the momentum equation, the
equation can be simplified for various applications. Approximations to the full dynamic wave
equations are accomplished by combining the continuity equation with the various
simplification of the momentum equation. The most common approximations of the
momentum equation are shown in Figure 7.6 (Henderson 1966). Although the time
derivative in Equation 7.7 is only included in the full dynamic wave equations, each of the
approximations can be used in unsteady flow analysis by coupling with the unsteady
continuity equation (Equation 7.2).
7.6
Sf
S0
y/x
(V/g)(V/x) -
(1/g)(V/t)
7.7
Cross
Section
Upstream
Limits
Reach
Bridge
Downstream
Limit
Stage
hydrograph. The magnitude of this loop can be affected by any of several hydraulic
parameters, the most significant of which may be backwater effects from downstream.
Each flood will follow a different loop. Figure 7.8 illustrates looped rating curves for two
floods with the arrows designating the rise and recession limbs of the discharge
hydrographs. The inner loop is for a slower rate of rise and fall which creates a narrower
loop.
Discharge
Figure 7.8. Looped rating curves showing the rising and recession limbs of a hydrograph.
3. The total flow downstream from a junction of a two tributaries is not necessarily the
combination of the two flows. Backwater from the flow at the junction can cause water to
be stored in upstream areas, reducing the flow combinations.
4. Tributary flows entering a main stream channel may experience a flow reversal caused
by flow in the main stem backing up into the tributary or vice versa e.g., when a large
tributary flood enters the main channel during a period of low flow.
5. If the inflow or stage at a boundary is changing rapidly, the acceleration terms in the
momentum equation are important and thus unsteady flow is a more robust and complete
computation. Examples of the phenomena are dam break analysis, rapid gate openings
and closures. Regardless of the slope of the channel, unsteady flow analysis should be
used for all rapidly changing hydrographs.
6. For full networks, where the flow divides and recombines, unsteady flow analysis should
be considered for subcritical flow. Unless the problem is simple, steady flow analyses
cannot accurately compute the flow distribution. When flow divides and recombines in
the split-flow reaches the length of the channels, the resistance to flow, and channel
geometry will differ. This causes the flood wave to travel through the reaches at different
speeds, which in turn affects the flow distribution in the reaches. To accurately determine
the flow distribution, unsteady flow modeling is preferred over steady state modeling.
7.9
(7.8)
where K represents any solution variable (e.g., velocity, discharge, flow depth, etc.).
7.10
j+ 2
j+ 1
K/x
K/t
j
x
i
i+ 1
i+ 2
K ij+1 K ij
x i
(7.9)
Variables other than derivatives are approximated at the time level where the spatial
derivatives are evaluated by using the same weighting factors, i.e.,
K j+1 + K ij+1
+ (1 )
K = i+1
K ij+1 + K ij
(7.10)
The influence of the weighting factor on the accuracy of the computations was examined
by Fread (1974), who concluded that the accuracy decreases as departs from 0.5 and
approaches 1.0.
This effect becomes more pronounced as the magnitude of the
computational time step increases. Usually, a weighting factor of 0.60 is recommended to
minimize the loss of accuracy while avoiding instability. When the finite difference operators
as defined in the three equations above are used to replace the derivatives and other
variables, the four-point implicit difference equations are obtained.
The terms associated with the jth time line are known from either the initial condition or
previous computations. The initial conditions are values of h and Q at each node along the x
axis for the first time line (j=1).
Since there are four unknowns and only two equations, the algebraic approximation of the
Saint-Venant Equations cannot be solved in an explicit or direct manner. However, if the
equations are applied to each of the N-1 rectangular grids between the upstream and
downstream boundaries (Figure 7.9), a total of 2N-2 equations with 2N unknowns can be
formulated (where N denotes the number of nodes). Then, prescribed boundary conditions,
one at the upstream boundary and one at the downstream boundary, provide the necessary
two additional equations required for the system to be determinate.
7.11
In order to solve the unsteady flow equations, the state of the initial conditions of water
surface elevation and discharge (h and Q) must be known at all cross sections at the
beginning (t = t0) of the simulation (represented by the squares in Figure 7.3). This is the
initial condition of the flow, which is typically steady nonuniform flow and would be solved
using HEC-RAS (steady) or WSPRO computer models.
When the flow is subcritical, information at both the upstream and the downstream boundary
of the system is also needed. The information supplied at a boundary is called a boundary
condition. This information can be in one of three forms: flow known as a function of time
(flow hydrograph), water-surface elevation known as a function of time (stage hydrograph),
or a relation between flow and water-surface elevation (rating curve or energy slope for river
conditions). The upstream boundary is typically a flow hydrograph (represented by the circles
in Figure 7.3) and the downstream boundary is typically a known relation between flow and
water-surface elevation (a rating curve or energy slope for river conditions). If the riverine
system is influenced by a tidal condition, then the downstream boundary condition is almost
always modeled using a stage hydrograph of tides.
The information supplied at a special feature internal to the stream system is often called an
internal boundary condition. In unsteady-flow analysis, internal boundary conditions are
approximated as steady-flow relations because the special features generally are short
enough that the changes in momentum and volume of water within the special features are
small. The isolation and description of the special features is a major component of
unsteady-flow analysis.
The same computational problems can arise for unsteady-flow analysis as for steady-flow
analysis because both analyses use algebraic approximations to the differential and integral
terms. These approximations are developed for a computational element of finite length. If
the computational element is too long, an incorrect solution results. The difference between
the analyses is that in unsteady-flow analysis the computational problems are more complex
and more frequent than in steady-flow analysis. The increased frequency is primarily
because unsteady-flow analysis involves computations over a wide range of water-surface
elevations, whereas most steady-flow analysis involves computations over a narrow range of
water-surface elevations. Furthermore, the time dimension results in additional
complications. Generally, the closer cross-sections are spaced, the shorter time-step is
required. Therefore the need to reduce cross-section spacing must be balanced with the
length of the simulation.
7.5 CHANNEL AND FLOODPLAIN CROSS SECTION
Figure 7.10 illustrates the interaction between the channel and the floodplain flows. When
the river is rising, flow moves laterally away from the channel, inundating the floodplain and
filling available storage areas. As the depth increases, the floodplain begins to convey water
downstream. When the river stage is falling, water moves back toward the channel from the
floodplains supplementing the main flow in the channel.
Even though the flow is two-dimensional, because the primary direction of the flow is along
the main channel, it can be approximated by a one-dimensional representation. Off-channel
ponding areas can be modeled as storage areas that exchange water to and from the
channel or to other storage areas within the floodplain. For this case, modeling the flow
using a steady state approximation will produce very different results than if modeled using
an unsteady approximation due to differences caused by storage.
7.12
Floodplain Boundary
Floodplain Flow
Floodplain Flow
Floodplain Boundary
i+ 1
7.13
For unsteady flow, when the water surface exceeds the trigger elevation, the ineffective flow
area will either begin to convey flow or remain ineffective depending upon its type
(permanent on non-permanent). For non-permanent ineffective flow areas, once the water
surface is higher than the trigger elevation, the entire ineffective flow area becomes effective.
Water is assumed to be able to move freely in that area based on the roughness, wetted
perimeter, and area of each subdivision. The left and right overbanks are no longer
considered storage but are now active flow areas.
Occasionally, ineffective flow areas may need to remain ineffective permanently. When the
water surface is below the trigger elevation, the permanent ineffective flow area behaves like
the non-permanent area. For permanent ineffective flow areas, when the water surface
elevation surpasses the trigger elevation, the area below the trigger elevation remains
effective. Water above the trigger elevation is assumed to convey flow. This option is useful
to avoid numerical instability associated with sudden change in conveyance.
7.6 STORAGE AREAS AND CONNECTIONS
Unsteady flow analyses are used to predict the temporal and spatial variations of a flood
hydrograph as it moves through a river reach. The effects of storage and flow resistance
within a river reach are reflected by changes in hydrograph shape and timing as the
floodwave moves downstream. Figure 7.11 shows the major changes that occur to a
discharge hydrograph as a floodwave moves downstream.
The HEC-RAS computer program provides an option to enter off-channel storage areas as
ponding areas that are either in-line or off-line. Storage areas are treated as simple
reservoirs.
The use of storage areas allow for more stable and faster computations than
representing a region with cross sections. The continuity equation is used to account for
volume of the storage area and the flow into and out of the storage is accomplished with the
storage indication method (i.e., as reservoir routing also sometimes called level pool routing).
The momentum equation is not computed and the storage is computed by volume/elevation
methods of either an area time relationship to account for depth or inputting an
elevation/volume curve. Figure 7.12 is an example of an off-line storage area used in the
HEC-RAS computer program.
Storage areas can be connected to a cross section(s) using a lateral connection, placed at
the top or bottom of a reach, or connected to another storage area. The only data needed to
describe storage areas are storage versus elevation. Two methods are available for this in
the program: surface area times depth, or interpolation from an entered rating curve of
elevation versus volume.
The data for storage area connections are a combination of procedures available in the
computer model. The storage area can be connected to river reach with a lateral connection
that can include gated structures and culverts, or entered as a stage-discharge rating curve.
An initial water surface elevation or a storage area is also required for simulation.
7.7 HYDRAULIC PROPERTY TABLES
The HEC-RAS computer program has several features that aid in the computation and
trouble shooting of the unsteady flow program for problems that may be encountered during
a computer run. The Geometric Preprocessor is one such feature. It is used to process the
geometric data into a series of hydraulic properties tables, rating curves, and a family of
rating curves. This is done in order to speed up the unsteady flow calculations. Instead of
calculating hydraulic variables for each cross section during each iteration the program
interpolates the hydraulic variables from the tables.
7.14
Travel Time
Discharge
Attenuation
Inflow
Hydrograph
at Point i
Upstream
Water
Entering
Storage
Routed Hydrograph at
Point (i+1) Downstream
Water
Leaving Storage
Time
Figure 7.12. Illustration of an off-line storage area using the HEC-RAS computed model.
7.15
Cross sections are processed into tables of elevation versus hydraulic properties of area,
conveyances, and storage (see Figure 7.13 for how the channel is subdivided). The user is
required to set the interval to be used for spacing the point in the cross section tables. This
interval is very important, in that it will define the limits of the table that is built for each cross
section. The interval must be large enough to encompass the full range of stages that may
be incurred during the unsteady flow simulations. On the other hand, if the interval is too
large, the tables will not have enough resolution to accurately depict the changes in area,
conveyance, and storage with respect to elevation. Another benefit of the hydraulic tables is
that they can be used to troubleshoot geometric problems that arise when running unsteady
flow models. The output from the geometric processor can be viewed either as hydraulic
property tables or plots of the rating curves. Viewing the graphical output is a useful
diagnostic tool for examining cross section geometry. The relationship of area, storage, and
conveyance should be examined for abrupt changes with elevation. Any abrupt change
should be reviewed to determine the overall significance for that particular run.
7.16
Figure 7.14 shows an anomaly in the conveyance for the right overbank flow at elevation
214.2. The cross-sectional plot shown in Figure 7.15 shows that the overbank Manning n
value is a constant for the right overbank having a value of 0.06. Due to the computational
scheme in HEC-RAS which subdivides the cross section by the number of horizontal n
values, a problem could exist in determining the proper conveyance. By adding another
Manning n value beginning at station 1024, the program will subdivide the cross section by
the two n values and recompute the conveyance. Note that when the program was rerun
with two n (but the same) values for the right overbank, the conveyance curve was much
smoother as shown in Figure 7.16. The feature of viewing these characteristic curves can
help the modeler troubleshoot many of the geometry properties that will cause the unsteady
flow computation to be unstable.
Elevation
Discontinuity
Figure 7.14. Conveyance properties versus elevation for a single cross section.
7.17
Elevation 214.2
Station = 1024
7.18
Figure 7.16. Conveyance properties versus elevation for a single cross section with
right overbank subdivided.
A numerical model is unstable when numerical errors grow to the extent at which the solution
begins to oscillate, or the errors become so large that the computations cannot continue.
Factors affecting model stability include: cross section spacing, computational time step,
theta weighting factor, solution iteration, and solution tolerances.
Rapidly rising hydrographs can cause computational problems, instability and nonconvergence, when applied to numerical approximations of the unsteady flow equations.
This is the case when an implicit, non-linear finite difference solution technique is used,
which is the case for most of the numerical solutions of the Saint-Venant equations.
However, many computational problems can be overcome with proper selection of the time
step t and the distance step x.
Cross section spacing should be placed at representative locations to describe changes in
geometry. Additional cross sections should be added at locations where changes occur in
discharge, slope, velocity, and roughness. Cross sections also should be added at
structures located along the river reach. Bed slope plays an important role in cross section
spacing. Steeper slopes require more sections, and streams flowing at high velocities also
will require more cross sections.
7.19
Computational time step is related to numerical stability and accuracy through the Courant
Condition.
Cr = Vw(t/x) 1.0 or t x/Vw
(7.12)
The flood wave speed is normally greater than the average velocity. For most rivers, the
flood wave velocity can be calculated by Vw = dQ/dA and an approximate value is Vw = 1.5V.
The Courant condition may yield time steps that are too restrictive (i.e., a larger time step
could be used and still maintain accuracy and stability. Fread (1981) found for many practical
unsteady flow problems that the Courant conditions can be relaxed and values greater than 1
yield satisfactory results.
The theta weighting factor is applied to the finite difference approximations when solving the
unsteady flow equations. Theoretically, theta can vary from 0.5 to 1.0. However a practical
limit is from 0.6 to 1.0. A theta of 1.0 provides the most stability, while a value of 0.6
provides the most accuracy. When choosing theta, there is a balance between accuracy and
computational robustness. Larger values of theta produce solutions that are more robust
and less prone to blowing up. Small values of theta, while more accurate, tend to cause
oscillations in the solution, which are amplified if there are large numbers of internal
boundary conditions.
At each time step derivatives are estimated, the equations are solved, and all of the
computation nodes then are checked for numerical error. If the error is greater than the
allowable tolerances, the program will iterate. The default maximum number of iterations in
the HEC-RAS program is set at 20. More iterations will generally improve the solution.
Within the HEC-RAS program two solution tolerances can be set or changed. The water
surface calculation is set to 0.02 feet and the storage area elevation solution is set at 0.05
feet. These default values should be acceptable for many river simulations. Making the
tolerances larger can reduce the stability of the solution, and making them smaller can cause
the program to go to the maximum number of iterations.
7.9 TWO-DIMENSIONAL UNSTEADY FLOW MODELS
The governing equations for two-dimensional unsteady flow (Saint-Venant equations) are
presented in Chapter 6. The equations in two dimensions include additional terms, such as
wind stress, that are difficult to represent in one-dimensional models.
Just as the one-dimensional unsteady flow solution is more complex than the onedimensional steady flow solution, the two-dimensional solution is much more complex than
the one-dimensional solution. The two-dimensional modeling approach is most appropriate
to calculate:
7.20
Boundary conditions are required throughout the simulation just as in the one-dimensional
modeling. They are applied along the flow boundaries of the solution, and are required to
eliminate the constants of integration when the governing equations are numerically
integrated to solve for U, V, and h in the interior domain. External boundary nodes along the
downstream end of the network are typically assigned a water surface elevation and
boundary nodes along the upstream end of the network are typically assigned flow or
discharge. The use of boundary condition specification removes either the depth, or one or
both of the velocity components from the computations.
Dynamic simulations are used to model situations where water levels, flow rates, and
velocities can change over time, such as an estuary where ocean tides influence the flow.
For tidal flow situations, starting a model at low tide usually will more quickly attain realistic
flow conditions throughout the model domain. This is somewhat similar to setting critical
depth as a downstream boundary condition for a one-dimensional model; the M2 curve will
converge more quickly than an M1 backwater curve. If good prototype tidal data is not
accessible, then one alternative is to access or generate synthesized harmonic tidal data.
Several software packages which will generate harmonic tidal data at most USGS station
locations are available.
7.10 TIDAL WATERWAYS
The HEC-25 (FHWA 2004, 2008) manuals provide guidance on tidal hydrology, hydraulics,
and coastal issues related to highways. Tidal waterways are defined as any waterway either
dominated or influenced by tides and hurricane storm surges. The first step in evaluation of
highway crossings is to determine whether the bridge crosses a river which is influenced by
tidal fluctuations (tidally affected river crossing) or whether the bridge crosses a tidal inlet,
bay or estuary (tidally controlled). The flow in tidal inlets, bays and estuaries is
predominantly driven by tidal fluctuations (with flow reversal), whereas the flow in tidally
affected river crossings is driven by a combination of river flow and tidal fluctuations.
Therefore, tidally affected river crossings are not subject to flow reversal, but the downstream
tidal fluctuation acts as a cyclic downstream control. Tidally controlled river crossings will
exhibit flow reversal.
Tidally affected river crossings are characterized by both river flow and tidal fluctuations.
From a hydraulic stand point, the flow in the river is influenced by tidal fluctuations which
result in a cyclic variation in the downstream control of the tail water in the river estuary. The
degree to which tidal fluctuations influence the discharge at the river crossing depends on
such factors as the relative distance from the ocean to the crossing, riverbed slope, crosssectional area, storage volume, and hydraulic resistance. Although other factors are
involved, the relative distance of the river crossing from the ocean can be used as a
qualitative indicator of tidal influence. At one extreme, where the crossing is located far
upstream, the flow in the river may only be affected to a minor degree by changes in tail
water control due to tidal fluctuations. As such, the tidal fluctuation downstream will result in
only minor fluctuations in the depth, velocity, and discharge through the bridge.
As the distance from the crossing to the ocean is reduced, again assuming all other factors
as equal, the influence of the tidal fluctuations increases. Consequently, the degree of tail
water influence on flow hydraulics at the crossing increases. A limiting case occurs when the
magnitude of the tidal fluctuations is large enough to reduce the discharge through the bridge
crossing. Because of the storage of the river flow at high tide, the ebb tide will have a larger
discharge and velocities than the flood tide.
7.21
Wind is a significant component of surge at a coastline. The U.S. Army Corps of Engineers
Storm Surge Analysis manual (USACE 1986b) indicates that wind is the greatest component
of storm surge and that the peak surge occurs in the area of maximum winds. Use of a wind
field as a two-dimensional boundary condition may be necessary to model some tidal
waterway conditions. In determining the forces on bridges, the properties of the flow that
have the greatest impact are the height of the water and its velocity (FHWA 2009c). Wind
fields can create waves that significantly affect the bridge structure and therefore need to be
analyzed in all river crossings that are tidally influenced.
The damage to highway bridges in recent hurricanes was due primarily to wave attack on
storm surge (see Chapter 10). The damage was caused as the storm surge raised the water
level to an elevation where larger waves could strike the bridge superstructure. Individual
waves produce both an uplift force and a horizontal force on the bridge decks. The
magnitudes of these forces depend on wave characteristics and on the inundation of the
deck. The magnitude of wave uplift force from individual waves can exceed the weight of the
simple span bridge decks. The total resultant force is able to overcome any resistance
provided by the typically small connections between the pile caps and bridge decks. The
decks begin to progressively slide, "bump," or "hop" across the pile caps in the direction of
wave propagation.
The buoyancy of the bridge decks caused by air pockets trapped under the bridge decks
contribute to the total force on the individual bridge decks when the deck is submerged, i.e.,
when the storm surge elevation exceeds the bridge deck elevation. However, bridge decks
that were elevated above the storm surge still-water elevation were still damaged in both
Hurricanes Ivan and Katrina by waves.
This conclusion, that wave loads were the primary cause of damage, is based on post-storm
inspections of the damaged bridges in combination with numerical model hindcasts of the
wave and surge conditions during the storms, some exploratory laboratory tests, and a
review of the related coastal engineering literature.
7.22
CHAPTER 8
BRIDGE SCOUR CONSIDERATIONS AND
SCOUR COUNTERMEASURE HYDRAULIC ANALYSIS
8.1 INTRODUCTION
The most common cause of bridge failures is from floods eroding bed material from around
bridge foundations. Scour is the engineering term for the erosion of soil, alluvium or other
materials surrounding bridge foundations (piers and abutments) by flowing water. The HEC18 and HEC-20 manuals (FHWA 2012b, 2012a) are the primary FHWA resources for
guidance on evaluating scour and stream instability at bridge crossings. Safe bridge design
must account for scour conditions that may occur over the life of the bridge. Scour is
greatest during flood events when flow velocity and depth is highest, but the event-related
scour is in addition to the long-term stream instability components of channel shifting,
aggradation, and degradation.
Each of the scour components discussed in this chapter should be considered during bridge
design. It is important for bridge engineers to recognize that these scour and stream
instability components be considered over the life of the bridge. No equations for predicting
scour are provided in this chapter because updated equations and procedures may be
incorporated into future versions of HEC-18 and HEC-20, and because every type of scour is
not discussed in this chapter.
The FHWA HEC-18 and HEC-20 manuals are the primary source of guidance and
procedures for incorporating scour and stream instability into safe bridge design. The
American Association of State Highway and Transportation Officials (AASHTO 2010) LRFD
Design Specifications includes the following statements, regarding the factors related to
scour and stream instability that should be considered in bridge design:
Evaluation of bridge design alternatives shall consider stream instability, backwater, flow
distribution, stream velocities, scour potential, flood hazards, tidal dynamics (where
appropriate) and consistency with established criteria for the National Flood Insurance
Program.
Studies shall be carried out to evaluate the stability of the waterway and to assess the
impact of construction on the waterway.
(Consider) whether the stream reach is degrading, aggrading, or in equilibrium.
(Consider) the effect of natural geomorphic stream pattern changes on the proposed
structure.
For unstable streams or flow conditions, special studies shall be carried out to assess the
probable future changes to the plan form and profile of the stream and to determine
countermeasures to be incorporated in the design, or at a future time, for the safety of the
bridge and approach roadways.
For the design flood for scour, the streambed material in the scour prism above the total
scour line shall be assumed to have been removed for design conditions.
Locate abutments back from the channel banks where significant problems with
ice/debris buildup, scour, or channel stability are anticipated.
Design piers on floodplains as river piers. Locate their foundations at the appropriate
depth if there is a likelihood that the stream channel will shift during the life of the
structure or that channel cutoffs are likely to occur.
8.1
The added cost of making a bridge less vulnerable to damage from scour is small in
comparison to the total cost of a bridge failure.
This is only a partial list of AASHTO's specifications and commentary related to scour and
stream instability. These topics are a significant aspect of safe bridge design, and are a
complex combination of hydrologic, hydraulic, fluvial-geomorphic, erosion, scour, sediment
transport, geotechnical, and structural considerations. The following sections describe scour
and stream instability processes, how to obtain data from hydraulic models for computing
scour, and numerical modeling of scour countermeasures.
8.2 SCOUR CONCEPTS FOR BRIDGE DESIGN
(2.1)
(1.5)
(0.9)
(0.3)
Embankment
During flood flows, water is conveyed in the river channel and in the floodplains adjacent to
the channel.
Figure 8.1 illustrates a representative bridge crossing and the flow
characteristics for a flood condition. The figure includes streamlines and velocity contours.
The more widely spaced streamlines are in the floodplain and divide the flow into 5 percent
increments. The closely spaced streamlines are in the channel and divide the flow into 10
percent increments of flow. The road embankments constrict the flow into the bridge
opening where flow velocities are highest. Upstream of the bridge constriction, where flow is
fully expanded in the floodplain, approximately 65 percent of the flow is in the channel and 35
percent is in the floodplains. In the bridge opening, approximately 90 percent of the flow is in
the channel and 10 percent is in the floodplain area between the channel banks and
abutments (setback area). Flow velocity is less than 5.7 ft/s (1.7 m/s) in the upstream
channel and 1.2 ft/s (0.37 m/s) in the upstream floodplains. This compares with velocities in
bridge opening as high as 8.8 ft/s (2.7 m/s) in the channel and 4.4 ft/s (1.34 m/s) in the
setback areas. The higher velocities in the bridge opening generate much higher shear
stresses and are much more erosive than the upstream flow velocities. In addition to the
increased velocities, bridge structural elements (piers and abutments) locally obstruct flow
and cause additional erosion at these locations.
8.2
Streamlines
Scour is a significant concern during extreme flood events and bridges should be designed to
withstand the scour produced by these events. Channel geometry, which includes
aggradation, degradation, channel shifting, and channel widening, also changes during the
life of a bridge. Therefore, potential for stream instability should be a part of safe bridge
design.
8.3 TYPES OF SCOUR
8.3.1 Contraction Scour
Contraction scour is a sediment imbalance process that occurs during floods when the
sediment supply from upstream is less than the sediment transport capacity in the bridge
opening. There are two sediment supply conditions for contraction scour; clear water and
live bed. Clear-water contraction scour occurs when the upstream flow velocity is insufficient
to transport bed material. The HEC-18 manual (FHWA 2012b) includes equations for
determining the critical velocity when bed material movement is initiated, which depends on
flow depth and particle size. Clear-water conditions occur for fine sediment sizes (sands and
fine gravel) only when flow velocity is small and for coarse sediment sizes (coarse gravel and
cobbles) even for relatively high velocity. Live bed conditions occur when there is sufficient
flow velocity to transport bed material upstream of the bridge. Very fine sediment (clay and
silt) is often not found in channel beds in significant amounts and does not generally play a
role in either clear-water or live-bed contraction scour. The water may be turbid due to
suspended transport of silt and clay, but is still considered as clear-water from the standpoint
of bed material transport.
For clear-water contraction scour, the flow velocity in the bridge opening is sufficient to move
bed material even though the upstream flow velocity is too low for bed material movement.
For live-bed contraction scour, the higher flow velocity in the bridge opening has a greater
capacity for transporting sediment that is the upstream flow velocity. In either case, there is
an imbalance between sediment supply and sediment transport capacity, and contraction
scour occurs. The channel bed erodes and lowers, thereby increasing the flow depth and
decreasing the flow velocity until the bed material transport capacity equals the supply from
upstream. The erosion process takes time so depending on the duration of the flood, the
ultimate scour may not be achieved. Accurate contraction scour calculations depend on
having accurate estimates of flow distribution at the approach and bridge cross sections.
Flow is divided into channel, left floodplain and right floodplain in the fully expanded flow
upstream of the bridge, and divided into channel, left setback (floodplain) and right setback
areas under the bridge. These subarea discharges control the contraction scour process.
Live-Bed Contraction Scour. Live-bed scour almost always occurs in river channels during
flood events. Exceptions to this expectation are boulder-bed and bedrock channels that are
not alluvial. Channels that have significant levels of diversion and/or flood control may also
not be live-bed because the channel forming flows no longer occur. Figure 8.2 includes a
plan and profile sketch to illustrate the flow variables for live-bed contraction scour. At the
approach section (cross section 1), the flow velocity in the river channel is high enough to
transport bed material. The total sediment transport in the approach channel depends on the
flow depth (y1), velocity (V1), discharge (Q1), width (W1), and sediment size (represented by
the median bed material particle size, D50). At the bridge section (cross section 2), floodplain
flow has entered the channel so the channel discharge (Q2), velocity, and sediment transport
capacity are greater than in the channel at the approach section. A hydraulic model includes
a surveyed cross section at the bridge so the flow depth in the model is a pre-scour depth
(y0). The channel width at the bridge section (W2) is often similar to the upstream width, but
8.3
may be wider or narrower. The bridge section channel may also be partially blocked by piers
or by abutments that encroach into the channel, which results in W2 less than W1. The livebed scour equation is presented in HEC-18 (FHWA 2012b). The equation yields the total
flow depth including scour (y2), and the scour is the difference between this depth and the
pre-scour flow depth (ys-c = y2 y0). Because it is assumed that bed material size is
consistent along the channel reach, bed material size is only used to determine whether or
not live-bed conditions exist and does not actually appear in the live-bed contraction scour
equation. For live-bed conditions, a functional relationship for contraction scour is:
ys-c = fn (y1, Q1, W1, Q2, W2, y0)
(8.1)
Larger amounts of contraction scour occur for greater differences between channel
discharge at the approach and bridge sections. Also, scour increases for narrower channel
widths at the bridge section. The worst case live-bed contraction scour occurs then the
bridge abutments and road embankments encroach into the channel and the entire floodplain
flow is conveyed in the constricted channel at the bridge opening. Live-bed contraction scour
decreases as the abutments are set back farther from the channel banks and when fewer or
narrower piers are located in the channel.
Clear-Water Contraction Scour. Clear-water contraction scour is expected in the setback
areas under a bridge. The fully expanded floodplain flow upstream of the bridge usually has
a low velocity and would not be expected to mobilize the granular floodplain materials.
Floodplains are also often comprised of cohesive materials and vegetated. Therefore,
although very fine particles (silts and clays) may be transported in suspension, there is little
potential for bed material transport or live-bed scour in floodplains. Flow velocity in the
setback area under the bridge is, however, often high enough to cause erosion. Clear water
scour is, therefore, an erosion process based on flow velocity and shear stress. Figure 8.3
includes a plan and profile sketch of the clear-water contraction scour variables. The
important variable at the approach section (section 1) is velocity (V1), but is only used to
determine whether the velocity is less than the critical velocity for bed material transport.
This comparison should be made if there is any uncertainty about whether the upstream flow
is transporting bed material. The channel as well as the setback areas could have clearwater contraction scour, but most often only the setback areas will. If there is a relief bridge
through the embankment on the floodplain, this opening will also typically have clear-water
contraction scour.
The clear-water contraction scour equation is a function of only the hydraulic conditions in a
particular subarea, not upstream conditions. These variables include discharge (Q), width
(W), and flow depth before scour (y0). Clear-water contraction scour occurs until the lowering
of the ground, which increases depth and decreases flow velocity, produces a non-eroding
velocity. The non-eroding velocity is a function of grain size (D50) for non-cohesive soils and
is a function of critical shear stress (c) for cohesive soils. The HEC-18 manual (FHWA
2012b) includes equations for clear-water contraction scour. As with the live-bed contraction
scour equation, the clear-water contraction scour equation yields a total depth including
scour (y2) and the predicted scour is the difference between this depth and the pre-scour
depth (ys-c = y2 y0). For clear-water conditions, a functional relationship for contraction
scour is:
(8.2)
8.4
Plan View
QFLOODPLAIN
Floodplain
Q1 W1
Channel
W2 Q2
QFLOODPLAIN
Floodplain
Approach
Section
Bridge
Section
Water Surface
V1 y1
y2
y0
Channel Bed
yS
1
Profile View
8.5
Plan View
Floodplain
W Q
Channel
W Q
W Q
Floodplain
Approach
Section
Bridge
Section
Water Surface
V1 < Vc
y0
y2
Floodplain or Channel
yS
Profile View
8.6
Vertical Contraction Scour (Pressure Scour). Pressure scour is another type of contraction
scour, but created by a vertical constriction rather than a horizontal constriction. It can occur
even if no horizontal constriction is present. Pressure scour may be either live-bed or clearwater depending on the upstream flow and sediment characteristics. Prediction of pressure
flow scour at an inundated bridge deck may be important for safe bridge design and for
evaluation of scour at existing bridges. An experimentally and numerically calibrated-formula
reference was developed by FHWA (2012c) to calculate pressure scour depth under various
deck inundation conditions. This formula is included in HEC-18 (FHWA 2012b). Figure 8.4
illustrates the flow characteristics at a fully submerged bridge deck. The depth in the area of
maximum scour is comprised of three components, which are hc (the vertically contracted
depth not including scour), ys (the scour depth), and t (the boundary layer thickness).
Pressure conditions can significantly increase total scour at a bridge because flow depth is
referenced to the bottom of the bridge deck rather than the water surface, and because the
boundary layer thickness is an additional scour component. Using the contraction scour, ys-c,
computed from the relationships represented in Equations 8.1 and 8.2, and referencing flow
depth to the bottom of the deck the functional relationship for vertical contraction scour is:
ys-vc = fn (ys-c, t)
(8.3)
scour can occur in clear-water or live-bed conditions. There are many factors that influence
the magnitude of pier scour. Hydraulic factors are velocity (V), depth (y), and angle of attack
() of the flow approaching the pier, but outside the influence of the pier. Pier shape (,
including circular, square, sharp, rounded-, or rectangular-nosed), pier width (a), and pier
length (L) also contribute significantly to the amount of pier scour. Complex pier geometries
that include pile groups, pile caps, and footings must also be considered when computing
pier scour. Sediment size (D50), density ( = s/ - 1) and gradation () are included in some
pier scour equations. Although pier scour may appear to be a relatively simple process, the
calculations are often cumbersome for all but the simplest cases. HEC-18 (FHWA 2012b)
includes several pier scour equations for various conditions. A functional relationship for pier
scour is:
ys-p = fn (V, y, , a, L, , D50, , )
(8.4)
Figure 8.5. The main flow features forming the flow field at a cylindrical pier
(NCHRP 2011a).
Abutment Scour. Scour occurs at abutments when the roadway embankment and abutment
obstruct the flow. Abutment scour is a type of local scour, but is related to contraction scour
because the embankment is the primary cause of flow constriction.
NCHRP (2011b) conducted an evaluation of abutment scour processes and prediction
methods. The conclusions and recommendations that pertain to abutment scour evaluation
and safe bridge design include:
Contraction scour should be viewed as the reference scour depth for calculating
abutment scour. Abutment scour should be taken as the product of the contraction scour
caused by flow acceleration through the constricted opening multiplied by a factor
accounting for large-scale turbulence. This approach would replace the current approach
for adding contraction scour to a separately computed abutment scour.
8.8
Abutments should be designed to have a minimum setback distance from the channel
bank of the main channel with riprap protection of the embankment and a riprap apron to
protect against scour. The setback distance should accommodate the apron width
recommended in HEC-23 (FHWA 2009a).
Two-dimensional models should be used on all but the simplest bridge crossings as a
matter of course.
y s a = fn( y 2 , A , y 0 )
(8.5)
Abutment scour can result in geotechnical failures of the embankment or channel bank
materials. Once the geotechnical failure depth is reached, scour will not increase in depth
but will progress laterally, potentially creating a free-standing abutment foundation that would
act more as a pier from the standpoint of scour.
Figure 8.6. Flow structure in floodplain and main channel at a bridge opening
(NCHRP 2011b).
8.9
Figure 8.7. View down at debris and scour hole at upstream end of pier.
8.10
Rectangular debris
Figure 8.8. Idealized flow pattern and scour at pier with debris.
105
Legend
EG PF 1
WS PF 1
Ground
100
Ineff
Bank Sta
Pier Debris
95
90
85
80
0
100
200
300
400
500
600
700
8.11
800
Vertical Instability. Vertical change includes aggradation and degradation resulting from a
long-term excess or deficit in sediment supply, and from degradation caused by headcutting.
Long-term trends in discharge also impact channel geometry because channels that convey
larger flows tend to be wider and deeper. If a channel consistently conveys more water than
it has historically, the channel will enlarge. This can occur due to increased runoff from
urbanization, from climate change, and many other causes. Bridge inspection files that
include repeat cross section measurements are useful in identifying aggradation and
degradation problems and trends. The sediment transport chapter (Chapter 9) includes the
discussion of sediment continuity and how sediment transport concepts can be used to
analyze aggradation and degradation when there is an imbalance of sediment supply and
transport capacity.
Headcuts occur when channel degradation progresses up the channel and are caused when
the downstream base level of a channel is lowered. Figure 8.10 shows a headcut that will
migrate upstream and through the bridge crossing during future runoff events. Features of a
headcut that can threaten a bridge include long-term degradation that persists after the
headcut has migrated upstream of the bridge, plunge pool when headcut is under the bridge,
and channel widening that occurs because bed lowering can destabilize channel banks.
Lateral Instability. Figure 8.11 shows progressive channel migration over a 72 year period at
a highway crossing. The channel banks were identified and traced from historic and recent
aerial photography. These banklines not only show trends of channel migration down valley
and across valley, but also variability in channel width through time. The channel migration
process includes erosion of the bank materials, bank geotechnical failures, transport of the
eroded and failed materials, and sediment accretion on the insides of bends (point bars).
Reviewing historic aerial photography is not only useful for identifying the potential for lateral
instability problems at a bridge, but can be used to make predictions of channel location
during the life of the bridge. These photo-comparison techniques are presented in HEC-20
(FHWA 2012a). As illustrated in Figure 8.12, a single flood can also cause extreme channel
migration and widening, which for some regions can present significant challenges for bridge
design.
8.12
1937
1969
1994
Image 2009
8.13
1975 Bankline
1975
1986
Figure 8.12. Channel widening and meander migration on Carson River near Weeks,
Nevada.
8.3.5 Evaluating Channel Instability
Vertical and lateral instability is often identified during bridge inspections, through channel
reconnaissance during bridge design, or through comparison of recent and historic aerial
photography. Hydraulic modeling and sediment transport analysis can also be used to
evaluate channel instability. As discussed in Chapter 9, sediment transport analysis can be
used to evaluate channel aggradation and degradation trends over the life of a bridge. Even
when sediment transport modeling is not performed, hydraulic models, especially twodimensional models, can provide insight into vertical and lateral channel instability potential.
Locations where channel flow velocity is much higher than up- or downstream may be prone
to bed or bank erosion. Models can be used to predict a future condition, but they can also
be used to evaluate potential future conditions by configuring the model for expected channel
changes.
Model results should never be interpreted without considering the river characteristics.
Geologic controls, sediment characteristics, vegetation characteristics, and manmade
features may counteract erosion that may be expected from reviewing model results. It is
important the channel reconnaissance be performed and that the hydraulic engineer
develops an understanding of a wide range of fluvial geomorphic processes and potential
channel response as discussed in HEC-20 (FHWA 2012a).
8.4 COMPUTING SCOUR
Each of the types of scour relies on hydraulic variables as input to the scour calculations.
These variables include velocity, depth, discharge, flow width, unit discharge, and flow
direction. The quality and accuracy of hydraulic modeling directly impact the accuracy of
scour calculations. If model geometry is inaccurate, bank stations are not correctly or
consistently defined, Manning n values are not accurate, or model assumptions are violated,
then the poor quality of the hydraulic input data used in scour calculations can result in
unreasonable and incorrect scour estimates.
The variables listed above all depend on the suitability of the hydraulic model to define flow
distribution. For pier scour, the velocity and depth upstream of the pier are required input.
For contraction scour, the amounts of flow in the channel relative to the floodplains both
upstream and in the bridge opening are required. Abutment scour depends on the same flow
distribution information as contraction scour, but also requires an estimate of flow
concentration adjacent to the abutment.
8.14
The rest of this section provides discussion of extracting the necessary hydraulic information
from one- and two-dimensional models. Recognizing that two-dimensional models provide
more accurate representations of the flow field and flow distribution, FHWA encourages the
use of two-dimensional modeling for all but the most straightforward bridge crossings.
8.4.1 One-Dimensional Models
Figure 8.13 shows the minimum number of cross sections for a one-dimensional bridge
hydraulic model. The Exit cross section is required to establish the downstream boundary
condition for the model. Contraction and abutment scour calculations require channel and
floodplain discharges at the Approach section and bridge crossing. In a HEC-RAS model,
the Crossing includes bridge and roadway geometry data placed between two cross sections
that are adjacent to the bridge and roadway. One-dimensional models can provide estimates
of hydraulic variables by computing incremental conveyance throughout the cross section
and distributing flow in proportion to conveyance.
CROSSING
RIGHT OVERBANK
LEFT OVERBANK
8.15
125
120
115
110
Vmax
Ymax
105
100
95
0
1000
2000
3000
4000
5000
Although these assumptions affect the entire flow distribution to some extent, the area where
there is the greatest error is near the abutment where much higher velocity and flow
concentration (unit discharge) are expected.
8.4.2 Two-Dimensional Models
Two-dimensional model results are shown graphically as velocity contours and vectors in
Figure 8.15. Contours of depth are also available as graphical output. The figure depicts a
complex flow situation where a highway crosses a channel and wide floodplain. There is a
long, main channel bridge, a shorter relief bridge on the floodplain (upper right corner of the
figure) and another relief bridge further along the embankment (not shown in the figure).
There is also a narrow railroad embankment, which has a main channel bridge and two relief
bridges, downstream of the wide highway embankment.
8.16
For pier scour calculations, point values of velocity and depth can be obtained at any
location. Flow direction can also be determined from the model output to estimate angle of
attack at a pier. Figure 8.15 also shows four flow lines. Flow lines, also called flux lines and
continuity lines, are used in two-dimensional models to compute the discharge through an
area. The flow lines in this figure are positioned to compute channel discharge at the bridge
opening and approaching the bridge upstream, overbank flow in the wide floodplain under
the main channel bridge, and total flow in the relief bridge. The area, length, average
velocity, and average depth can also be determined from the flow line output. These
variables provide the input data for contraction and abutment scour calculations.
Figure 8.15 also shows the flow concentration (high velocity) at the two abutments of the
main channel bridge. This type of flow concentration is not available output from onedimensional models. Unit discharge can be computed at any point in the two-dimensional
model by multiplying velocity and depth, or at any flow line by dividing discharge by width
(flow line length). Although this is a much more accurate representation of flow than a onedimensional model, two-dimensional models also make simplifying assumptions, which
include hydrostatic pressure and no vertical velocity components.
Flow Lines
(2.4)
(1.8)
(1.2)
0.6)
(0.0)
8.17
Area of scour
REVETMENT ARMOR
(ROCK RIPRAP OR EQUAL)
QUARTER
ELLIPSE
VEGETATIVE
COVER
Ls
0.4 Ls
As shown in Figure 8.17, the geometry of guide banks can be included directly in twodimensional models. The finite element mesh shown in this figure demonstrates that areas
of rapid change in velocity magnitude or direction require a more refined network of
elements. The unstructured mesh of the finite element network also allows for detailed
assignment of cover type, i.e. Manning n. Figure 8.18 shows the flow field around this guide
bank and the flow around the abutment at the other end of the bridge. There is flow
separation under the bridge right abutment (left side of figure) but not on the guide bank side.
Flow velocities are also much lower at the guide bank protected side.
8.19
(3.0)
(2.4)
(1.8)
(1.2)
(0.6)
(0.0)
8.20
2 (0.6)
3
4
1 (0.3)
4
3
2
1
(a)
Velocity, ft/s, (m/s)
6 (1.8)
Spur 4
5 (1.5)
Spur 3
Spur 5
4 (1.2)
Spur 2
3 (0.9)
Spur 1
Spur 6
2 (0.6)
2
4
1 (0.3)
Spur 7
1
3
2
1
(b)
Figure 8.19. Two-dimensional analysis of flow along spurs, (a) flow field without spurs, and
(b) flow field with spurs.
8.21
8.22
CHAPTER 9
SEDIMENT TRANSPORT AND ALLUVIAL CHANNEL CONCEPTS
9.1 INTRODUCTION
Safe bridge design includes the recognition that channels are not stationary, but that they
may adjust their bed and banks during the life of the bridge. The HEC-20 (FHWA 2012a)
and HDS 6 manuals (FHWA 2001) are the primary FHWA manuals related to stream
instability and sediment transport topics. Another reference that provides broad coverage of
this topic is Sedimentation Engineering (ASCE 2008). HDS 6 states that "The moveable
boundary of the alluvial river adds another dimension to the design problem and can
compound environmental concerns. Therefore, the design of highway crossing and
encroachments in the river environment requires knowledge of the mechanics of alluvial
channel flow." This chapter provides an overview of these topics in the context of bridge
design.
Most channels and floodplains that roads cross are alluvial. Alluvial channels are formed by
materials that have been transported and deposited by flowing water and can be transported
by the channel in the future. Channel adjustments include aggradation, degradation, width
adjustment, and lateral shifting. Aggradation and degradation are the overall raising or
lowering of a channel bed over time from sediment accumulation or erosion. Channel
widening and shifting are the result of bank erosion due to hydraulic forces or by mass failure
of the bank.
Sediment transport analyses can play a role in several aspects of safe bridge design. Of
primary concern is whether the channel will experience long-term aggradation or
degradation. Aggradation decreases flow conveyance and has the potential of increased
frequency and magnitude of flooding, road overtopping, and loss of service. Degradation
threatens bridge foundations by removing support and making the bridge more vulnerable to
scour during floods. A related concern is that the bridge could alter the prevailing flow
conditions and cause aggradation or degradation. Departments of Transportation may also
conduct channel restoration as part of a bridge replacement. Sediment transport analysis is
needed to determine the potential impacts of the restoration to avoid creating a channel that
does not adequately convey sediment supplied from upstream. Another role that sediment
transport can play in bridge design is that contraction scour can be computed from a
sediment transport model rather than from the standard contraction scour equation. This
would be done if there was significant uncertainty in the use of the standard contraction
scour equation or if there was a significant potential benefit from applying a more detailed
analysis. In summary, sediment transport analyses should be considered as part of a bridge
design for the following reasons.
ASCE (2008) indicates that one-dimensional sediment transport models are most often
applied to simulations involving extended river reaches and extended time periods, typically
to determine the long-term response of a river to natural or man-made changes. This is
because of the computational efficiency of one-dimensional models as compared to twodimensional models. This makes one-dimensional models well-suited to address the topics
9.1
listed above. As indicated by ASCE (2008), one-dimensional models cannot resolve local
details of flow and mobile bed dynamics, which two- or three-dimensional models provide the
possibility of resolving, though currently for relatively small scale problems over relatively
short time periods.
Channel stability and sediment transport are complex processes that interact to produce the
existing channel form and future channel adjustments. This is why HEC-20 (FHWA 2012a)
emphasizes that qualitative evaluation (Level 1), and standard engineering analyses (Level
2) should be conducted even when advanced numerical sediment transport modeling (Level
3) is performed. Factors that influence sediment transport include sediment properties,
hydrology, watershed and land-use conditions, channel geometry, and vegetation.
Sediment properties include size, gradation, cohesion, density, shape, porosity of the
sediment mixture, angle of repose, and sediment layer depths. Many, if not all, aspects of
hydrology also play a role in sediment transport analyses. These include not only peak flow
rates, but also individual flood hydrographs, and the durations of all flows. The entire range
of flow may be significant because even though the highest flows have the highest rates of
sediment transport, lower flows may have significantly longer durations and produce the
greatest cumulative sediment transport. Channels respond and adjust to changes in flow
and sediment supply. Therefore, changing watershed conditions often result in adjustments
in channel geometry. Channel geometry, bed material, and vegetation determine hydraulic
variables (velocity, depth, etc.), which in turn control sediment transport capacity.
Consequently, sediment transport and channel stability depend not only on the specific
physical processes, but also the history of natural and human-induced factors in the
watershed.
The following sections provide a general overview of sediment transport concepts and
processes. Other resources are available to provide the in-depth information required to
perform these analyses. These resources include HDS 6 (FHWA 2001), Sedimentation
Engineering (ASCE 2008), textbooks (Simons and Senturk 1992, Yang 2003, Julien 2010),
and the manuals for specific numerical models that incorporate sediment transport.
9.2 SEDIMENT CONTINUITY
The amount of material transported, eroded, or deposited in an alluvial channel is a function
of sediment supply and channel transport capacity. Sediment supply is provided from the
tributary watershed and from erosion occurring in the upstream channel bed and banks.
Sediment transport capacity is primarily a function of sediment size and the hydraulic
properties of the channel. When the transport capacity of the flow equals sediment supply
from upstream, a state of equilibrium exists.
Application of the sediment continuity concept to a channel reach illustrates the relationship
between sediment supply and transport capacity. The sediment continuity concept states
that the sediment inflow minus the sediment outflow equals the rate of change of sediment
volume in a given reach. More simply stated, during a given time period the amount of
sediment coming into the reach minus the amount leaving the downstream end of the reach
equals the change in the amount of sediment stored in that reach (Figure 9.1). The sediment
inflow to a given reach is defined by the sediment supply from the watershed and channel
(upstream of the study reach plus lateral input directly to the study reach). The transport
capacity of the channel within the given reach defines the sediment outflow. Changes in the
sediment volume within the reach occur when the total input to the reach (sediment supply)
is not equal to the downstream output (sediment transport capacity). When the sediment
supply is less than the transport capacity, erosion (degradation) will occur in the reach so
that the transport capacity at the outlet is satisfied, unless controls exist that limit erosion.
9.2
Conversely, when the sediment supply is greater than the transport capacity, deposition
(aggradation) will occur in the reach.
Qs (in)
Z
X
Qs (out)
X
Y
Change in volume = Sediment inflow Sediment outflow
(erosion if negative, deposition if positive)
Q s
Z
=
t
X
(9.1)
where:
W
Z
t
Qs
X
=
=
=
=
=
=
t(Q s( in ) Q s( out ) )
V
=
WL (1 )
WL (1 )
(9.2)
where:
V
L
=
=
Change in volume of sediment particles stored or eroded in the reach, ft3 (m3)
Reach length, ft (m)
9.3
0 = ySo
(9.3)
where:
0
=
=
=
S0
Another useful formula for estimating average shear stress for gradually varied flow
conditions is:
nV
0 = (1/ 3 )
y
Ku
(9.4)
where:
n
V
Ku
Ku
=
=
=
=
Equation 9.4 shows the relationship between velocity and shear stress; shear stress is
proportional to velocity squared. The Shields parameter relates critical shear stress to
particle size and specific weight by.
c = k sD s ( s )
(9.5)
where:
c
ks
Ds
s
=
=
=
=
Shields parameter ranges from 0.03 to 0.10 for natural sediments and depends on particle
shape, angularity, gradation and imbrication. The use of 0.047 is common for sand sizes.
When the shear stress of the flow exceeds the critical shear stress of the particle, the
channel bed begins to mobilize and bed material is transported downstream. Particle motion
begins as sliding and rolling of individual particles along the bed. It is important to recognize
that the Shields equation is not a sediment transport equation because it does not provide
any estimate of the amount of sediment in motion. It is also important to note that only the
shear stress acting on the particles, or grain friction, should be used in applying this
relationship.
9.3.2 Modes of Sediment Transport
Once the critical shear stress is exceeded, bed material begins to move (roll, slide, and
saltate) along the bed surface. This material is referred to as bed load or contact load
because it is in almost continuous contact with the bed. For small amounts of positive
excess shear stress (defined as o - c), this is the only mode of bed material transport. As
excess shear stress increases, turbulence begins to suspend some of the particles. The
turbulence acts to mix the particles in the water column and gravity causes the particles to
settle. Therefore, bed material can also transported downstream as suspended bed material
load. The two types of bed material load are illustrated in Figure 9.2.
9.5
The suspended bed material load shown in Figure 9.2 is a result of the interaction between
gravity and turbulence. Because gravity is causing particles to settle, they are concentrated
near the bed. Turbulence mixes the particles in the water column and, depending on the
size and density of the particles, relatively few particles may reach the surface. The
suspension of particles is illustrated in Figure 9.3, which shows the concentration profile for
various particle sizes in a turbulent flow field. The equation that describes the concentration
profiles is:
y y a
c
= o
c a y y o a
(9.6)
where:
c
ca
y0
z
=
=
=
=
=
=
v*
=
=
=
=
g
R
1.0
Surface
0.9
1/64
0.8
1/32
0.7
0.6
1/16
(y-a)/(yo-a) 0.5
1/8
0.4
1/4
0.3
1/2
0.2
1
0.1
a/yo = 0.05
2
4
0.0
0.0
Bottom
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Larger particles have greater fall velocities and larger Rouse numbers. Therefore, Figure 9.3
shows that for a given level of turbulence (as represented by the shear velocity), large
particles will remain close to the bed. Finer particles have smaller Rouse numbers, are
mixed higher into the flow and have higher concentrations. Julien (2010) indicates that
particle sizes with Rouse numbers less than 0.025 (1/40) will have essentially uniform
concentration profiles. These particles are extremely fine, primarily silts and clays, and have
very small fall velocities. They are defined as wash load, which are derived primarily from
upland erosion and bank erosion of floodplain materials. Wash load material is not found in
appreciable quantities in the channel bed.
In summary, bed material is transported in contact with the bed (bed load) and in suspension
(suspended bed material load). The total sediment load transported by the channel also
includes wash load, which is supplied to the channel rather than derived from the bed. Wash
load is also transported in suspension. In coarse bed channels, such as cobble-bed and
boulder-bed streams, sand may act as wash load because it is not found in appreciable
quantities in the bed and because the supply is far less than the channel capacity to
transport this size.
9.3.3 Bed-Forms
In sand-bed streams, sand material is easily eroded and is continually being moved and
shaped by the flow. The interaction between the flow of the water-sediment mixture and the
sand-bed creates different bed configurations which change the resistance to flow, velocity,
water surface elevation and sediment transport. Consequently, an understanding of the
different types of bed forms that may occur, bed form geometry, resistance to flow, and
sediment transport associated with each bed form can help in analyzing flow in an alluvial
channel.
Flow Regime. Flow in alluvial sand-bed channels is divided into two regimes separated by a
transition zone. Forms of bed roughness in sand-bed channels are shown in Figure 9.4.
There is no direct relationship between the classification of upper and lower flow regime and
Froude Number (supercritical/subcritical flow). The flow regimes are:
The lower flow regime, where resistance to flow is large and sediment transport is small.
The bed form is either ripples or dunes or some combination of the two. Water surface
undulations are out of phase with the bed surface, and there is a relatively large
separation zone downstream from the crest of each ripple or dune. The velocity of the
downstream movement of the ripples or dunes depends on their height and the velocity
of the grains moving up their backs.
The transition zone, where the bed configuration may range from that typical of the lower
flow regime to that typical of the upper flow regime, depending mainly on antecedent
conditions. If the antecedent bed configuration is dunes, the depth or slope can be
increased to values more consistent with those of the upper flow regime without changing
the bed form; or, conversely, if the antecedent bed is plane, depth and slope can be
decreased to values more consistent with those of the lower flow regime without
changing the bed form. Resistance to flow and sediment transport also have the same
variability as the bed configuration in the transition zone. This phenomenon can be
explained by the changes in resistance to flow and, consequently, the changes in depth
and slope as the bed form changes.
The upper flow regime, in which resistance to flow is small and sediment transport is
large. The usual bed forms are plane bed or antidunes. The water surface is in phase
with the bed surface and normally the fluid does not separate from the boundary, except
when an antidune breaks.
9.7
Figure 9.4. Bed forms in sand channels (after HDS 6 - FHWA 2001).
Effects of Bed Forms at Stream Crossings. At high flows, most sand-bed stream channels
shift from a dune bed to a transition or a plane bed configuration. The resistance to flow is
then decreased by one-half to one-third of that preceding the shift in bed form. The increase
in velocity and corresponding decrease in depth may increase scour around bridge piers,
abutments, spur dikes or banks and may increase the required size of riprap.
Another effect of bed forms on highway crossings is that with dunes on the bed, there is a
fluctuating pattern of scour on the bed. Methods for computing bed-form geometry can be
found in Julien and Klaassen (1995) and Karim (1999). Karim included laboratory and field
data where the crest-to-trough height, , for dunes ranged from less than 0.1y to up to 0.5y.
Karim also showed a range of antidune heights between 0.1y and 0.4y. Bennet (USGS
1997) indicated an approximate upper limit as < 0.4y. The average dune height equation
by Julien and Klaassen is:
= 2.5 50
y
y
0 .3
(9.7)
The lower and upper bounds on dune heights (95 percent) range from 0.3 to 3.2 times this
average height. Dune lengths can be approximated as 6.25 times the flow depth. Care must
be used in analyzing crossings of sand-bed streams in order to anticipate changes that may
occur in bed forms and the impact of these changes on the resistance to flow, sediment
transport, and the stability of the reach and highway structures. With a dune bed, the
Manning n could be more than twice as large as a plane bed (see Figure 9.5). A change
from a dune bed to a plane bed, or the reverse, can have an appreciable effect on depth and
velocity. In the design of a bridge or a stream stability or scour countermeasure, it is good
engineering practice to assume a dune bed (large n value) when establishing the water
surface elevations, and a plane bed (low n value) for calculations involving velocity.
9.8
BED FORM
Plane Bed
Dunes
Ripples
Transition
Plane Bed
Standing waves
and antidunes
Water
Surface
Bed
Resistance to flow
(Mannings roughness
Coefficient)
Lower Regime
Upper Regime
Transition
STREAM POWER
Figure 9.5. Relative resistance to flow in sand-bed channels (after USGS 1989).
9.4 OVERVIEW OF SEDIMENT TRANSPORT EQUATIONS AND SELECTION
Equations for predicting bed material sediment transport differ depending on the mode of
sediment transport. ASCE (2008) includes 16 bed load equations. The Meyer-Peter and
Mller (1948) equation is considered to be a classic bed load equation. The equation has
the basic form of:
qb = a( 0 c ) 3 / 2
(9.8)
where:
qb
a
=
=
As with the analysis of incipient motion, only grain friction should be included in the bed
shear (0) variable. Many of the equations presented in ASCE (2008) include excess shear
stress (0 c) to the 1.5 power. Because bed shear is proportional to velocity squared (see
Equation 9.4), bed-load dominated sediment transport, such as in gravel-bed rivers, is
generally proportional to velocity cubed.
Another classic method for predicting sediment transport is the Colby (1964) graphical
method for bed material load in sand-bed rivers. The Colby method is discussed in detail in
HDS 6 (FHWA 2001). Sand-bed channels are dominated by suspended sediment transport
for most flow conditions. The first step in the Colby method is to determine an uncorrected
sediment discharge based on flow velocity. The Colby curves follow a trend of sediment
discharge proportional to velocity to the power of between 3.5 and 6. These large powers
indicate that suspension is more effective in transporting sediment in sand-bed channels.
They also indicate that uncertainty in velocity generates extreme uncertainty in sediment
9.9
qs =
vcdy
(9.9)
=
=
The solution of the integral uses Equation 9.6 for sediment concentration and a logarithmic
velocity profile equation (vertical velocity distribution is discussed in Chapter 6). The
concentration and velocity profiles are illustrated in Figure 9.6. This integration depends on a
reference concentration that is determined from the bed load. ASCE (2008) presents nine
equations for determining the reference concentration and an easily applied equation (Abad
and Garcia 2006) to solve the integration of Equation 9.9. Because the rate of bed load
transport and the concentration profile depend on grain size, the integration is performed for
the range of grain sizes in the bed material and the total bed material load is the sum of the
proportionate transport rates computed for each size class. Julien (2010) used Equation 9.9
to show that bed load comprises 80 percent or more of the total load when shear velocity
divided by fall velocity (v*/) is less than 0.5, and that suspended load comprises 80 percent
or more of the total load when v*/ > 2. For 0.5 < v*/ < 2 the sediment transport is
considered to be mixed load.
ASCE (2008) also presents six empirically based equations for determining total sediment
load. These equations have the advantage of being more easily applied, but should only be
used within the limits of the data used in their development. This concept applies to the use
of any sediment transport equation. The HDS 6 manual (FHWA 2001) includes 20 sediment
transport equations and the applicability to various grain sizes. The HEC-RAS Reference
Manual (USACE 2010c) and SAM reference manual (USACE 2002) include information on
the range of data (particle size, specific gravity, velocity, depth, slope, channel width and
temperature) used to develop many of the sediment transport equations used for sand and
gravel sizes. Any equation that is considered for use should be evaluated for applicability to
the specific conditions.
9.10
9.11
The next level of complexity is SIAM. SIAM is a sediment budget tool. It combines channel
reach-weighted hydraulics, annual flow duration information, bed material gradation, other
sediment properties, and information on sediment sources to compute the annual sediment
transport capacity for each reach. The engineer must identify channel reaches with similar
hydraulic and sediment properties. The results can be used to locate potential instabilities
and sediment imbalances (surpluses and deficits) between reaches.
The third, and most challenging capability is sediment transport routing. In sediment routing,
the sediment transport capacity is used to update cross section geometry, which is then used
to update the hydraulic calculations. The geometry is updated for individual cross sections,
though the hydraulic variables can be weighted with up- and downstream cross sections. A
flood hydrograph or long-term flow hydrograph is entered as a series of constant flows.
Within each flow time step, many sediment transport and cross section updating time steps
are often required. The model does not assume that transport capacity is reached at every
cross section, but limits erosion based on potential entrainment rates and limits deposition
based on fall velocity, flow velocity and water depth. Sediment layer depths, as well as
lateral limits for erosion and deposition are also input. Sediment transport modeling also
requires greater model upstream and downstream extent, as well as careful consideration of
all boundary conditions (hydraulic and sediment).
Figure 9.7 shows channel profiles for Las Vegas Wash, a channel with a history of
degradation. The channel has experienced increased flow over time and sediment supply is
limited by upstream channel stabilization. The bridge crossing location degraded over 30 ft
(9 m) between 1970 and 1999. Equilibrium slope calculations indicated that an additional 40
feet (12 m) of degradation could occur based on expected discharge and sediment supply
rates. Equilibrium slope is defined as the slope a channel will seek based on an expected
combination of sediment supply and water discharge, and is described in detail in the HEC20 manual (FHWA 2012a). Because equilibrium slope calculations do not provide the
amount of time it will take to reach equilibrium, a sediment transport model was developed to
provide an independent estimate of channel degradation at the bridge and the time it would
take to reach various amounts of degradation. The final profile from the sediment transport
model is at equilibrium with the expected flow and sediment and was achieved approximately
10 years into the simulation. The bridge was protected with grade control structures and the
predicted final degradation has occurred downstream of the bridge. Note that the final profile
shows aggradation downstream of station 6000 ft (1830 m). This aggradation is due to
sediment accumulating in the pool of Lake Mead, which is a downstream, though highly
variable, control.
Because of the sensitivity to the hydraulic conditions, a sediment transport routing model will
often highlight deficiencies in a hydraulic model. When velocity or conveyance change
significantly between cross sections, the change in sediment transport capacity may result in
unrealistic amounts of aggradation or degradation, or create unrecoverable numerical
instabilities during the model run. Sediment transport routing is inherently non-uniform and
unsteady. It is non-uniform because the cross section geometry will change as erosion and
deposition occur. It is unsteady because the rate of sediment transport imbalance
determines the amount of cross section change (Equations 9.1 and 9.2).
9.12
1280
(390)
1260
(384)
1240
(378)
1220
(372)
1200
(366)
1180
(360)
Bridge Crossing
Location
1160
(354)
1140
(347)
0 (0)
2500 (760)
5000 (1520)
7500 (2290)
10000 (3050)
12500
(3810)
There are many decisions that impact the results of a bridge hydraulic analysis. Selecting
high Manning n values results in conservative water surface elevations. Selecting low
roughness values results in more conservative velocity estimates. Using a fixed bed model
for bridge hydraulics may also produce conservative estimates of backwater. This is
because contraction scour enlarges the bridge opening and reduces the velocity in the
bridge. Therefore, in many cases, backwater actually caused by bridges is less than a fixed
bed model predicts. In some cases, use of a mobile-bed model, or incorporating contraction
scour in the bridge opening of a fixed-bed model, better represents actual flow conditions at
the bridge. This is illustrated in Figure 9.8, which shows the water structure for a fixed-bed
model run for natural (no bridge) and bridge conditions and a mobile-bed model. The bed
profile shows the construction scour caused by the bridge constriction. In this case, the
mobile-bed model computed approximately 40% less backwater due to the 2.6 ft (0.79 m) of
contraction scour that resulted based on sediment transport.
9.13
250
500
1000
1250
1017
1500
310
Road Embankment
309
1013
308
1009
Elevation (ft)
306
1001
305
997
304
993
Elevation (m)
307
1005
302
989
301
985
300
981
0
1000
2000
3000
4000
299
5000
Figure 9.8. Contraction scour and water surface for fixed-bed and mobile-bed models.
9.6 ALLUVIAL FANS
Alluvial fans are very dynamic sedimentary landforms that can create significant hazards to
highways as a result of floods, debris flows, deposition, channel incision, and avulsion
(Schumm and Lagasse 1998). They can occur where there is a change from a steep to a flat
gradient, especially in mountainous regions. The National Research Council Committee on
Alluvial Fan Flooding (NRC 1996) defined alluvial fans as sedimentary deposits that are
convex in cross-profile and located at a topographic break, such as the base a mountain,
escarpment, or valley side, that is composed of stream flow and/or debris flow sediments and
that has the shape of a fan either fully or partially extended. As the bed material and water
reaches the flatter section of the stream, the coarser bed materials can no longer be
transported because of the sudden reduction in both slope and velocity. Consequently, a
cone or fan builds out as the material is dropped. Alluvial fans are often characterized by
unstable channel geometries and rapid lateral movement. The steep channel tends to drop
part of its sediment load in the main channel building out into the main stream. In some
instances, the main stream can make drastic changes, or avulsions, during major floods.
The NRC committee determined that alluvial fan hazards can include (1) flow path
uncertainty below the fan apex, (2) abrupt deposition and ensuing erosion of sediment as a
stream or debris flow loses competence to carry material eroded from the steeper, upstream
source area, and (3) the combination of sediment availability, slope, and topography creates
ultra hazardous conditions that elevation or fill will not reliably mitigate risk.
9.14
The potential for avulsion, deposition, channel blockage, and channel incision are important
for highway design. To minimize these impacts on highways, a reconnaissance of the fan
and its drainage should be undertaken so that potential changes can be identified and
countermeasures taken. Any study of alluvial fans should include a geomorphic map
delineating active and inactive portions of the fan and the identification of problem sites
within the active portions of the fan. For example, local aggradation in a channel can lead to
avulsion because avulsion is likely to occur in places where deposition has raised the floor of
the channel to a level that is nearly as high as the surrounding fan surface. This condition
can be identified in the field by observation or by the surveying of cross-fan profiles (Schumm
and Lagasse 1998).
French (1987) cautions that alluvial fan hydraulics are highly unsteady and two-dimensional.
Analyzing hydraulic and sediment transport conditions on alluvial fans should not be
conducted without in-depth geomorphic evaluation. ASCE (2008) indicates that there are
two-dimensional models available for modeling flow and sediment transport on alluvial fans,
specifically FLO-2D (Obrien 2009). FLO-2D is a grid-based finite difference model that is
well-suited for simulating unconfined flow and sediment conditions that occur on alluvial fans,
including mud- and debris-flow conditions. Although the grid-based approach is less suited
for determining the detailed hydraulic results often desired for bridge applications, the highly
unsteady, unconfined flow conditions on alluvial fans are the dominant processes and make
this approach necessary.
9.15
9.16
CHAPTER 10
OTHER CONSIDERATIONS
10.1 HYDRAULIC FORCES ON BRIDGE ELEMENTS
10.1.1 General
Bridge design engineers must analyze the stability of the bridge as a whole and elements of
the bridge under various loading conditions. Rivers, streams and coastal water bodies exert
significant forces on bridge structures especially during times of flood or storm surge. The
hydraulic forces potentially acting on a bridge include hydrostatic, buoyancy, drag and wave
forces. Impact by vessels and forces exerted by debris or ice are also closely tied to
hydraulics. Bridge designers require information from the results of the hydraulic analysis to
evaluate the hydraulic forces on bridge elements.
Bridge designers typically follow the AASHTO LRFD Bridge Design Specifications (herein
referred to as the LRFD Specifications) (AASHTO 2010), with some state-specific
modifications, in evaluating forces and loads on bridges. The guidelines of LRFD
Specifications, along with information and insights from other references, are briefly
summarized in this section.
10.1.2 Hydrostatic Force
The weight of water exerts hydrostatic pressure in all directions. It is calculated as the
product of the height of the water surface above the point of interest and the unit weight of
water. Thus the pressure is greatest at the lowest point of a submerged element and is zero
at the water surface elevation.
The hydrostatic force acting on a bridge element in a particular direction is the summation, or
integral, of the product of the pressure and the surface area of the bridge element projected
in the plane perpendicular to the direction of the force. Hydrostatic forces on one side of a
bridge are at least partly balanced by opposing hydrostatic forces acting on the other side.
Any imbalance in the hydrostatic force is due to variation in the water surface elevation.
Bridge designers must be informed of the water surface elevation upstream and downstream
of the bridge for the design flood in order to evaluate the hydrostatic forces.
10.1.3 Buoyancy Force
Buoyancy is an uplift force equivalent to the weight of water displaced by the submerged
element. It can be a threat to a submerged bridge superstructure if the superstructure design
incorporates large enclosed voids as with a box-girder or if air pockets develop between
girders beneath the deck. Buoyancy is also a factor in evaluating wave-related forces on
bridge decks, discussed later in this chapter. If a pier is constructed with a large empty void,
the buoyant uplift force acting on the pier may be significant. Bridge designers must be
informed of the water surface elevation upstream and downstream of the bridge for the
design flood in order to evaluate the buoyancy forces.
10.1
Figure 10.1. CFD results plot showing velocity direction and magnitude from a model
of a six-girder bridge (from FHWA 2009c).
10.2
The resulting report, entitled "Hydrodynamic Forces on Bridge Decks" (FHWA 2009c)
provides equations for use in determining the drag coefficient, lift coefficient and moment
coefficient as functions of the inundation ratio, for each of the three superstructure types
investigated. The inundation ratio is a measure of the degree of submergence of the
superstructure. It is defined as the vertical distance measured down from the water surface
to the bridge low chord divided by the depth of the superstructure measured vertically from
the top of the parapet to the low chord. For the six-girder superstructure, the equations yield
drag coefficients roughly ranging from 0.7 to 2.2.
For inundated bridge decks, lift is another force component that should be considered in
bridge design. FHWA (2009c) provides equations for lift, as well as the resulting turning
moment that the combined drag and lift forces create. The deck may not react as a single
unit depending on the interconnection of the girders, so lift and drag may be more severe for
individual deck elements. Figure 10.2 is a definition sketch for drag, lift, and turning moment
variables.
Figure 10.2. Definition sketch for deck force variables (FHWA 2009c).
Equations for computing drag, lift, and moment per unit length of bridge (FHWA 2009c) are:
h* =
hu hb
s
(10.1)
FD =
CD V 2 s
; for h * > 1
2
(10.2)
FD =
C D V 2 s h *
; for h * < 1
2
(10.3)
FL =
C L V 2 W
2
(10.4)
Mcg =
C M V 2 W 2
2
(10.5)
10.3
The values of the drag, lift and moment coefficients for a six-girder bridge are shown in
Figures 10.3 through 10.5. FHWA (2009c) also provides charts of these coefficients for
three-girder bridges.
2.5
1.5
Exp. Fr=0.16
CD
Exp. Fr=0.22
Exp. Fr=0.28
Exp. Fr=0.32
Fitting Equation
0.5
STAR-CD
Fluent LES
Fluent k-epsilon
0
0
0.5
1.5
2.5
3.5
h*
0.5
CL -0.5
Exp. Fr=0.16
Exp. Fr=0.22
Exp. Fr=0.28
Exp. Fr=0.32
Fitting Equations
STAR-CD
Fluent LES
Fluent k-(epsilon)
-1
-1.5
-2
0
0.5
1.5
2.5
h*
10.4
3.5
0.35
Exp. Fr=0.16
0.3
Exp. Fr=0.22
0.25
Exp. Fr=0.28
Exp. Fr=0.32
0.2
STAR-CD
Fluent LES
0.15
Fluent k-(epsilon)
CM
0.1
0.05
0
-0.05
-0.1
-0.15
0
0.5
1.5
2.5
3.5
h*
10.5
10.6
the Froude number in the contracted section. The report also provides useful guidance on
the selection of the reference velocity for use in the drag force, or stream pressure,
calculations. The hydrostatic force is calculated based on the difference in water surface
from the upstream side of the debris accumulation to the downstream side of the bridge.
Another research project by the NCHRP used field observations, a photographic database
and extensive physical modeling to investigate the affects of debris on bridge pier scour. The
resulting report, titled "Effects of Debris on Bridge Pier Scour," (NCHRP 2010b, Report 653)
provides refined guidance on estimating the potential dimensions of a debris flow blockage,
on incorporating debris into one- and two-dimensional hydraulic models, and on computing
an effective pier width for pier scour calculations based on the estimated debris dimensions.
When the potential for debris accumulation on the bridge is significant, the hydraulic engineer
should be prepared to provide the bridge designer with the estimated dimensions and
reference elevation of the potential debris blockage. The hydraulic engineer should also
recommend an appropriate drag coefficient for the debris, based on NCHRP Report 445.
10.1.7 Effects of Ice
When ice accumulates at a bridge and forms a substantial ice jam, significant problems can
develop. Some of the negative consequences include bridge scour and bank erosion, even
during times of low streamflow. Ice jams also impart significant lateral forces on the bridge.
Similar to debris blockages, ice jams magnify the stream pressure forces by increasing the
surface area to which the stream pressure is applied. The upstream water surface elevation
(and consequently the hydrostatic force) is affected by the inordinate amount of backwater
that often accompanies ice jams. The elevation at which ice is expected to accumulate has a
significant influence on the bridge stability calculations. Extensive discussion on evaluation of
ice forces is provided in the LRFD Specifications.
The design team should perform site-specific research to assess whether ice jamming is a
relevant concern. If it is a concern, the hydraulic engineer may be required to develop
hydrologic and hydraulic information to assist the bridge designer in evaluating ice forces. It
may be beneficial, for instance, to determine the months of the year when ice jamming is
most likely to occur. Streamflow records would then be studied to assess the potential for
flooding during the most likely ice jamming months, and to identify a streamflow rate that
represents a reasonable yet conservative flow rate for assessing the potential elevation of an
ice jam on the bridge. Field reconnaissance may reveal evidence of the elevation range
within which ice jams typically form. The Transportation Association of Canada has published
the "Guide to Bridge Hydraulics" (TAC 2004), which includes information on estimating the
stage and thickness of ice jams. If needed, the hydraulic engineer can develop hydraulic
model simulations of ice jam situations. The HEC-RAS program includes the capability to
incorporate ice cover into its simulations.
Ice can exert other forces on a bridge besides the increase in stream pressure and
hydrostatic force mentioned above. Large ice floes striking bridge piers can generate
significant impact forces. Large sheets of ice can experience thermal expansion, generating
lateral pressure on the bridge. Ice adhering to the bridge structure during water level
increases can impart uplift forces. The hydraulic engineer should be prepared to assist the
bridge designer in assessing the potential range of water levels associated with these forces.
10.7
with the piers making only a small contribution. The relatively small backwater contribution of
piers, however, can be a significant factor in bridge design, especially in the context of highly
restrictive, no-rise floodway regulations (see Chapter 2).
Most bridges crossing regulatory floodways require piers to be located within the floodway to
keep the span lengths within a cost effective range and to avoid unreasonable superstructure
depths. The placement of piers in the floodway, however, can lead to regulatory challenges
since the piers are likely to cause some small amount of backwater. In such cases the
hydraulic engineer and the design team should work together to develop a design for the
spans and piers that satisfies the regulatory constraints without unacceptable cost impacts.
Aligning the piers with the flow direction, using hydraulically streamlined pier geometries and
elevating the low chord to a reasonable freeboard height above the 100-year flood elevation
are best practices that should be incorporated to the extent feasible. In such cases the bridge
hydraulics should be analyzed using a methodology that accounts for the hydraulic benefits
of streamlined pier geometries. The momentum method within HEC-RAS, for instance (see
Chapter 5) uses a pier drag coefficient that is a function of the pier geometry. Yarnell's
equation also incorporates a pier shape factor.
The Texas Department of Transportation commissioned a research study to aid in the
evaluation of the magnitude and nature of backwater associated with bridge piers. The
researchers conducted physical modeling to correlate the pier drag coefficient and the
relative backwater (backwater depth divided by flow depth) to the Froude number
downstream of the pier, for a range of pier sizes and flow contraction ratios. The results of
the study (Charbeneau and Holly 2001) led to a recommended equation for calculating the
backwater effects of pier. The recommended equation follows the form of Yarnell's equation
but incorporates modifications for improved correlation to the physical modeling results. In
general it was found that the observed relative backwater depth was consistently less than
Yarnell's equation would predict.
Another analysis strategy that can be useful in dealing with no-rise floodway regulations is to
perform a simulation that includes only the bridge elements that are actually located within
the floodway, excluding elements of the crossing that are outside the floodway. This
simulation allows the hydraulic engineer to isolate the impacts caused by work in the
floodway. Only work in the floodway is regulated to the no-rise standard per FEMA
regulations, though local ordinances may regulate to a no-rise standard outside the FEMA
regulatory floodway.
10.3 COINCIDENT FLOWS AT CONFLUENCES
10.3.1 Significance of Coincident Flows at Confluences
When a bridge over a stream is located near a confluence with another stream, the engineer
must consider the potential influence of the other stream on the hydraulics at the crossing.
Questions to consider include:
If the bridge is upstream of the confluence: How will the other stream affect the water
surface profile through the bridge waterway for various flood recurrence intervals?
If the bridge is within or very near the floodplain confluence zone: How will the interaction
between the flows from the two streams affect the distribution and direction of flow
throughout the confluence area?
10.9
In order to appropriately consider the effects of the confluence, it is necessary to estimate the
coincident flow probabilities of the two streams. Consider a bridge crossing a minor tributary
stream a short distance upstream from a major river, as illustrated in Figure 10.7. The
tributary would likely have a much smaller contributing drainage area than the river. Major
flooding on the river may be driven by different factors than those that cause major flooding
on the tributary. Floods on large rivers, for instance, are often driven by spring runoff
supplemented by long-duration spring rainfall, and may last for weeks. Floods on smaller
tributaries are often driven by intense thunderstorms at other times of year, and typically last
a few hours.
NCHRP conducted a research project (Project 15-36) NCHRP (2010c) with the objective of
developing practical, reliable procedures for estimating coincident flow probabilities at
confluences. The work has been completed and at present publication is pending. The
resulting report, once published, is expected to provide significantly improved guidance on
handling the issue of coincident flows at confluences.
10.4 ADVANCED BRIDGE HYDRAULICS MODELING
10.4.1 Background
One-dimensional and two-dimensional hydraulic analysis techniques are sufficiently rigorous
for the needs of most bridge design projects. Occasionally, however, a project calls for more
advanced hydraulic modeling techniques, such as physical modeling or computational fluid
dynamics (CFD) modeling, which do not require the simplifications inherent in one- and twodimensional modeling.
Physical hydraulic modeling refers to simulations conducted in a geometrically-scaled
physical representation of the bridge or, more often, an element or section of the bridge
along with the surrounding channel or waterway. CFD modeling refers to a highly detailed
three-dimensional mathematical representation of the bridge element and waterway. Both
techniques allow for the investigation of flow patterns and hydrodynamic phenomena at a
degree of resolution, detail and rigor that is not readily obtainable with one-dimensional or
two-dimensional analysis.
10.4.2 Applications of Advanced Modeling
Physical Modeling. Numerous physical model studies have been conducted to assess scour
potential in situations involving large piers with complex geometry. Physical modeling has
also been used to evaluate vertical contraction scour as illustrated in Figure 10.8. HEC-18
(FHWA 2012b) provides recommended equations for estimating the scour potential at
complex piers, but their range of reliable application does not cover the full range of possible
complex configurations. Physical modeling, therefore, is sometimes used to enhance the
reliability of the scour estimates. Physical modeling provides the benefit of demonstrating,
resolving and displaying the complex flow behavior without reliance on numerical
formulations that are, of necessity, only approximate representations of the real physical
conditions. Physical modeling also allows a more detailed understanding of the geometric
configuration of scour around the pier.
Physical modeling for scour investigations is conducted in moveable-bed flumes. The flumes
are constructed as geometrically scaled models of the prototypes they represent. Most of the
limitations of physical modeling stem from the challenge of scaling the hydraulic conditions
from the prototype to the model. Hydraulic scaling for bridge hydraulics applications is
usually based on the Froude number, meaning that the Froude number in the model is set to
equal the Froude number in the prototype under design conditions. Even with Froude
number scaling, challenges can arise which are described later in this section. To support
physical modeling, two-dimensional computer modeling is often conducted to determine the
velocity magnitude and direction and the depth of flow at each pier in the prototype for the
design flow conditions.
10.11
Figure 10.8. Velocity from physical modeling using Particle Image Velocimetry (PIV).
CFD Modeling. CFD modeling employs numerical methods to solve the Navier-Stokes
equation in analyzing detailed three-dimensional fluid flow patterns for a wide range of
applications, from stream hydraulics to aircraft design to medical studies of flow through
blood vessels. CFD modeling is able to resolve complex near-field flow patterns, such as the
vortices in the vicinity of flow obstructions, provided the grid cells of the model are properly
sized and configured. Figure 10.9 is an example of CFD modeling of a submerged bridge
deck. The general applicability of CFD to bridge hydraulics is, to date, somewhat limited.
When CFD is applied to bridge hydraulics, it is usually directed toward local scour prediction
or the analysis of hydraulic forces on bridge piers and superstructures.
10.12
(a)
(b)
(c)
10.13
(d)
Three-dimensional CFD modeling of flow and scour at complex piers with large dolphins
Large-scale and small-scale physical modeling of scour at complex piers with large
dolphins and fenders
The CFD modeling was conducted using an enhanced version of the CCHE3-D software.
The modeling combined 3-dimensional flow dynamics and fully coupled sediment transport
(Dou et al. 2001). The work focused on scour simulations at a limited number of channel
piers, and the model domain for the simulation of each pier was limited to the near vicinity of
the pier, as shown in Figure 10.11.
Figure 10.11. Illustration of CFD modeling of Woodrow Wilson Bridge pier and dolphins
(from Dou et al. 2001).
10.14
The large-scale tests at the Turner Fairbank Highway Research Center were much more
costly to conduct. Consequently only four large-scale experiments were performed. The
large-scale tests had model-to-prototype scale ratios of 1 to 28 for one test and 1 to 50 for
the other three. The purpose of the large-scale models was to investigate the scale effects by
comparison with small-scale models of the same conditions. The comparison between largeand small-scale models showed enough similarity to provide confidence in the use of the
small-scale tests to predict the scour at the piers (Jones 2000). The physical models
provided significant value in comparing the effects of different design options. They showed,
for example, that the use of three 45-foot diameter dolphins to protect the bascule piers from
vessel collision could double the scour potential at the piers, while the use of an alternate
fender ring could actually reduce the scour potential. Figure 10.12 is a photograph of one of
the experiments at the J. Sterling Jones Hydraulics Research Laboratory.
10.15
Technical challenges in physical modeling are typically related to scaling between the model
and the prototype. The most common type of scaling for open-channel hydraulic studies is
Froude-based scaling, which generally means that the geometric configuration is scaled
down by some uniform scale ratio, but hydraulic properties have different scale ratios such
that the Froude number in the model is the same as in the prototype. Froude scaling is a
reasonably straightforward way to establish similitude in models of open-channel flow.
Unfortunately, the scaling of sediment sizes and sediment transport in a physical model are
not straightforward for Froude scaling. Recognizing this limitation, physical model studies for
scour evaluation often use a bed material size that has a critical velocity that is just less than
the model velocity, rather than attempting to scale the sediment size from the prototype.
The scaling requirements of physical modeling lead to another limitation in bridge hydraulic
modeling. Unless the depth to width ratio is distorted in the model, it is usually not feasible to
physically model the entire bridge waterway and floodplain unless the available flume facility
is uncommonly wide. Most physical model studies applied to bridge hydraulics, therefore, are
designed to represent the flow around and adjacent to a specific bridge element, such as a
pier or abutment. Supplemental two-dimensional computer modeling is often employed in
order to apply the correct local velocity and flow direction in the physical model.
Limitations of CFD Models. CFD modeling is not yet in widespread use for bridge hydraulics.
Practical limitations of CFD are associated with the required amount of computational
resources and the limited availability of personnel qualified to develop and apply CFD
models. The examples to date of CFD being applied to bridge hydraulics problems required
the use of very powerful computers, which are not widely available. As a result of the
computational intensity, most CFD studies applied to bridge hydraulics have focused on a
specific local flow field, for instance at a pier, rather than attempting to model the entire
bridge waterway. Therefore CFD is usually a supplement to, rather than a substitute for, oneor two-dimensional modeling. As with physical modeling, the number of personnel with
expertise in CFD modeling, especially as applied to bridges, is relatively small.
The current technical limitations of CFD in bridge hydraulics, as with physical modeling,
relate to sediment transport and scour processes. The Woodrow Wilson Bridge example
cited above required the simplifying assumption of uniform sand, where the actual bed
material was varied and included cohesive soil. Significant refinement is required to the
computational algorithms of CFD models if they are to be validated for direct use in
predicting scour depths.
10.5 BRIDGE DECK DRAINAGE DESIGN
10.5.1 Objectives of Bridge Deck Drainage Design
The design of a bridge should include consideration of bridge deck drainage in order to
protect public safety, support efficient traffic flow and prevent or minimize water related
damage to the bridge. Relevant design measures include the use of appropriate cross
slopes and longitudinal slopes on the bridge deck, along with hardware such as inlets,
scuppers, and drainage pipes. While the concerns and design approaches are comparable to
roadway pavement drainage design, significant differences exist because of the physical and
geometric constraints of installing a drainage system on a bridge.
FHWA document HEC-21 "Design of Bridge Deck Drainage," (FHWA 1993) provides
extensive guidance on the design of deck drainage systems. This section briefly summarizes
the design considerations for bridge deck drainage, drawing heavily from HEC-21.
10.16
Flow switching from the gutter on one side of the road to the other side
A sag in the gutter profile, causing water to pond
A locally flattened cross slope allowing excessively wide flow spread
10.17
Protecting Road Embankments at Bridge Ends. Erosion damage commonly occurs on the
road embankment slopes adjacent to bridge ends because of inadequate control of bridge
deck drainage. The problem can be minimized by designing the drainage system to deliver
the flow safely to the bottom of the embankment without erosion. HEC-21 advises
intercepting gutter flow with roadway drainage inlets on the approaches at both ends of the
bridge. The intercepted flow is typically either conveyed by pipes to the bottom of the
embankment or is delivered through pipes to an existing storm drain system. Some flow can
be allowed to bypass the specified roadway inlet if there is curbing of sufficient height and
length to convey the bypassed flow to the next drainage inlet or erosion-protected outfall
location, thus protecting the embankment from erosion damage.
Minimizing Drainage-Related Damage at Bridge Joints. Water seeping from the deck
through bridge joints can cause corrosion damage to the girders, bearings and substructure.
For that reason, some transportation agencies require inlets on the bridge deck to capture
flow before it runs across the joint at the downslope end of the bridge, even if there are no
other inlets on the deck. The interception capacity of a single bridge deck inlet is quite
limited. Therefore the beneficial function of bridge deck inlets placed at the downslope end of
a bridge deck is primarily to intercept nuisance flows such as runoff from minor rainfall
events, snowmelt, and landscape watering, rather than to keep joints dry during highintensity rainfall events.
10.5.3 Design Rainfall Intensity
The runoff flow rate that must be accommodated in bridge deck drainage design is directly
related to the short-duration rainfall intensity, which is the expected temporal rate of rainfall
over a brief period of time (usually 10 minutes or less). Transportation agencies typically link
the criteria for acceptable spread width and protection of the embankments at the bridge
ends to a standard design recurrence interval (frequency) for rainfall intensity. The 10-year
rainfall intensity is commonly used as the design standard for moderate- to high-volume
roads. HEC-22 (FHWA 2009b) provides guidance on selecting design rainfall frequency for
deck drainage.
10.5.4 Practical Considerations in Design of Bridge Deck Inlets and Drainage Systems
Dimensional Limitations of Bridge Deck Inlets. The pavement drainage inlets used in
roadway pavement drainage applications are generally unsuitable for bridge deck
applications because they cannot be easily integrated into the structural dimensions of a
bridge deck. Roadway pavement drainage typically drops through a long curb opening or
gutter grate into a large concrete catch basin, from which it is discharged through a pipe into
an outfall or a storm drain system.
Bridge deck inlets, by necessity, usually have a smaller footprint on the bridge deck surface.
Large openings may cause extensive complications in the design and construction of deck
reinforcement. Bridge deck inlets are typically rectangular or round cast iron grates that allow
runoff to drop into shallow inlet chambers constructed of formed concrete, ductile iron or
welded steel. HEC-21 provides illustrations of several common inlet configurations, and also
explains the factors that affect the interception capacity of bridge deck inlets. Grates with
bars parallel to the traffic direction are the most hydraulically efficient. Many new bridges
and bridge widenings, however, are being designed to accommodate bicycle traffic. Such
bridges require bicycle-safe grates, which have bars perpendicular to the traffic direction.
Perpendicular-bar grates can be made more efficient with vane grates, which are tilted or
curved with the top edges inclined in the upstream direction.
10.18
Handling Intercepted Runoff. From the shallow inlet chambers, the drainage is discharged
either into a vertical scupper or an underdeck drainage system. A vertical scupper may
discharge drainage water directly into the air under the bridge or may extend down to the
ground along the height of a pier. In many situations the runoff cannot be discharged directly
into receiving waters beneath the bridge due to storm water quality concerns or regulations.
In such cases any water intercepted from the bridge deck must be conveyed in an underdeck
drainage system to a point on the stream bank or shore, where the drainage can then be
discharged to an underground storm drain system or to an appropriate storm water quality
feature. Direct discharge of drainage into the air below the bridge can also be restricted for
other reasons. Roads, railroads, and residential, commercial or industrial development
beneath the bridge are examples of settings in which direct discharge from the bridge deck is
unacceptable.
As a general rule, underdeck drainage systems are problematic to the construction,
maintenance and aesthetics of bridges, and should be avoided unless they are required by
the setting or by regulations. If they cannot be avoided, they should be kept as short as
possible. Underdeck drainage pipe is usually ductile iron, PVC or fiberglass and is typically
of smaller diameter than the conduit used in underground storm drains. Figure 10.14 is a
photograph of an installed underdeck drainage system, constructed of fiberglass pipe.
10.19
Water Quality Impacts on Receiving Waters. When a bridge is to cross a wide waterway, the
design of bridge deck drainage can be a significant challenge due to the need to avoid
negatively impacting the water quality of the waterway being crossed. When environmentally
sensitive waters are present, the state's water quality regulations will prohibit direct discharge
of the bridge deck runoff into the stream beneath the bridge. In such cases the runoff must
be captured and conveyed off of the bridge to an acceptable stormwater quality mitigation
feature (termed a stormwater best-management practice, or BMP). The NCHRP report titled
"Assessing the Impacts of Bridge Deck Runoff Contaminants in Receiving Waters" is a
resource to aid in identifying, assessing and managing the water quality aspects of bridge
deck runoff (NCHRP 2002).
Maintenance Considerations. Even under the best conditions, bridge deck inlets tend to
become plugged by debris. To minimize the required maintenance effort and promote the
efficiency of the bridge deck drainage system, inlets and under deck bridge drainage
systems should be designed to keep debris at or above the bridge deck surface, and should
be located in areas that are easy to reach and safe for maintenance crews to service. Inlets
should be placed at the outer edge of the shoulder, and the shoulder should be as wide as
feasible. Inlets should not be located within traffic lanes, unless current and projected traffic
volumes are very low.
10.20
CHAPTER 11
LITERATURE CITED
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11.2
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11.3
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11.7
11.8
APPENDIX A
Metric System, Conversion Factors, and Water Properties
A.1
A.2
APPENDIX A
Metric System, Conversion Factors, and Water Properties
The following information is summarized from the Federal Highway Administration, National
Highway Institute (NHI) Course No. 12301, "Metric (SI) Training for Highway Agencies." For
additional information, refer to the Participant Notebook for NHI Course No. 12301.
In SI there are seven base units, many derived units and two supplemental units (Table A.1).
Base units uniquely describe a property requiring measurement. One of the most common
units in civil engineering is length, with a base unit of meters in SI. Decimal multiples of
meter include the kilometer (1000m), the centimeter (1m/100) and the millimeter (1m/1000).
The second base unit relevant to highway applications is the kilogram, a measure of mass
which is the inertial of an object. There is a subtle difference between mass and weight. In
SI, mass is a base unit, while weight is a derived quantity related to mass and the
acceleration of gravity, sometimes referred to as the force of gravity. In SI the unit of mass is
the kilogram and the unit of weight/force is the newton. Table A.2 illustrates the relationship
of mass and weight. The unit of time is the same in SI as in the U.S. Customary system
(seconds). The measurement of temperature is Centigrade. The following equation converts
Fahrenheit temperatures to Centigrade, C = 5/9 (F - 32).
Derived units are formed by combining base units to express other characteristics. Common
derived units in highway drainage engineering include area, volume, velocity, and density.
Some derived units have special names (Table A.3).
Table A.4 provides useful conversion factors from U.S. Customary to SI units. The symbols
used in this table for metric units, including the use of upper and lower case (e.g., kilometer
is "km" and a newton is "N") are the standards that should be followed. Table A.5 provides
the standard SI prefixes and their definitions.
Table A.6 provides physical properties of water at atmospheric pressure in SI system of
units. Table A.7 gives the sediment grade scale and Table A.8 gives some common
equivalent hydraulic units.
A.3
System
U.S. Customary
SI
A.4
Name
hertz
newton
pascal
joule
watt
coulomb
volt
farad
ohm
siemens
weber
tesla
henry
lumen
lux
A.5
Symbol
Hz
N
Pa
J
W
C
V
F
S
Wb
T
H
lm
lx
Expression
s-1
kg m/s2
N/m2
Nm
J/s
As
W/A
C/V
V/A
A/V
Vs
Wb/m2
Wb/A
cd sr
lm/m2
Quantity
Length
Length
Length
Length
Area
Area
Area
Area
Area
Area
Volume
Volume
Volume
Volume
Volume
Volume
Volume
Mass
Mass
Mass/unit length
Mass/unit area
Mass density
Force
Force
Force/unit length
Force/unit length
Pressure, stress, modulus of elasticity
Pressure, stress, modulus of elasticity
Pressure, stress, modulus of elasticity
Pressure, stress, modulus of elasticity
Bending moment, torque
Bending moment, torque
Moment of mass
Moment of inertia
psf
pcf
lb
kip
plf
klf
psf
ksf
psi
ksi
ft-lb
ft-kip
lb ft
lb ft2
A.6
kg/m2
3
Multiply
by *
1.609
0.9144
0.3048
25.40
2.590
4047
0.4047
0.8361
0.09290
645.2
1233
0.7646
0.02832
28.32
0.2360
3.785
16.39
0.4536
0.4536
1.488
4.882
kg/m
N
kN
N/m
kN/m
Pa
kPa
kPa
MPa
Nm
kN m
m
kg m2
16.02
4.448
4.448
14.59
14.59
47.88
47.88
6.895
6.895
1.356
1.356
0.1383
0.04214
mm4
mm3
kW
kW
W
W
416200
16390
3.517
1.054
745.7
0.2931
Quantity
Multiply by *
ft3/s
m3/s
0.02832
cfm
cfm
mgd
m /s
L/s
m3/s
0.0004719
0.4719
0.0438
ft/s
m/s
0.3048
m/s
kg m/s
0.3048
0.1383
Angular momentum
lb ft2/s
kg m2/s
Plane angle
degree
rad
Plane angle
degree
mrad
*4 significant figures; underline denotes exact conversion
0.04214
0.01745
17.45
f/s
lb ft/sec
Submultiple
Factor
10-1
10-2
10-3
10-6
10-9
10-12
10-15
10-18
10-21
10-24
Submultiple
Symbol
d
c
m
n
p
f
a
z
y
Multiple
Name
deka
hecto
kilo
mega
giga
tera
peta
exa
zetta
yotto
A.7
Multiple
Factor
101
102
103
106
109
1012
1015
1018
1021
1024
Multiple
Symbol
da
h
k
M
G
T
P
E
Z
Y
Density
Specific
weight
Dynamic
Viscosity
Kinematic
Viscosity
Vapor
Pressure
Surface
Tension1
Bulk
Modulus
m2/s
N/m2 abs.
N/m
GN/m2
611
0.0756
1.99
872
0.0749
2.05
1,230
0.0742
2.11
1,700
0.0735
2.16
2,340
0.0728
2.20
3,170
0.0720
2.23
4,250
0.0712
2.25
5,630
0.0704
2.27
7,380
0.0696
2.28
12,300
0.0679
blank
20,000
0.0662
blank
31,200
0.0644
blank
47,400
0.0626
blank
70,100
0.0607
blank
101,300
0.0589
blank
Centigrade
Fahrenheit
kg/m3
N/m3
N.s/m2
32
1,000
9,810
1.79 x 10-
41
1,000
9,810
-3
1.51 x 10
-3
10
50
1,000
9,810
1.31 x 10
15
59
999
9,800
1.14 x 10
20
68
25
77
996
9,790
997
-3
-3
1.00 x 10
-4
7.20 x 10-
9,732
-4
8.53 x 10
-4
50
122
988
9,693
5.47 x 10
60
140
983
9,843
4.68 x 10-
9,694
4.04 x 10
-6
1.00 x 10
-7
8.94 x 10
-7
6.58 x 10
-7
5.53 x 10
4.74 x 104.13 x 10
3.64 x 10
3.26 x 10
176
972
9,535
3.54 x 10
194
965
9,467
3.15 x 10-
-4
2.82 x 10
-7
-4
90
9,398
-4
80
958
9,751
212
1.14 x 10-
7.24 x 10-
9,771
994
100
1.31 x 10
996
978
-6
95
158
1.51 x 10
8.00 x 10-
86
70
-6
35
992
8.91 x 10
30
104
1.79 x 10-
9,781
7.97 x 10-
40
-7
-7
-7
2.94 x 10
Table A.7. Physical Properties of Water at Atmospheric Pressure in U.S. Customary Units.
Temperature
Density
Specific
Weight
Dynamic
Viscosity
Kinematic
Viscosity
Vapor
Pressure
Surface
1
Tension
Bulk
Modulus
Weight
Ib/ft3
Ib-sec/ft2
4
x 10-
ft2/sec
-5
x 10
Ib/in2
Ib/ft
Ib/in2
Fahrenheit
Centigrade
Slugs/ft
32
39.2
40
50
60
70
80
90
100
120
140
160
180
200
0
4.0
4.4
10.0
15.6
21.1
26.7
32.2
37.8
48.9
60.0
71.1
82.2
93.3
1.940
1.940
1.940
1.940
1.939
1.936
1.934
1.931
1.927
1.918
1.908
1.896
1.883
1.869
62.416
62.424
62.423
62.408
62.366
62.300
62.217
62.118
61.998
61.719
61.386
61.006
60.586
60.135
0.374
blank
1.93
blank
0.09
blank
0.00518
blank
287,000
blank
0.323
0.273
0.235
0.205
0.180
0.160
0.143
0.117
0.0979
0.0835
0.0726
0.0637
1.67
1.41
1.21
1.06
0.929
0.828
0.741
0.610
0.513
0.440
0.385
0.341
0.12
0.18
0.26
0.36
0.51
0.70
0.95
1.69
2.89
4.74
7.51
11.52
212
100
1.847
59.843
0.0593
0.319
14.70
0.00514
0.00508
0.00504
0.00497
0.00492
0.00486
0.00479
0.00466
blank
blank
blank
blank
blank
296,000
305,000
313,000
319,000
325,000
329,000
331,000
332,000
blank
blank
blank
blank
blank
A.8
Millimeters
4000-2000
2000-1000
1000-500
500-250
250-130
130-64
64-32
32-16
16-8
8-4
4-2
2-1
1-1/2
1/2-1/4
1/4-1/8
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
2.00-1.00
1.00-0.50
0.50-0.25
0.25-0.125
2000-1000
1000-500
500-250
250-125
1/8-1/16
1/16-1/32
1/32-1/64
1/64-1/128
0.125-0.062
0062-0031
0.031-0.016
0.016-0.008
125-62
62-31
31-16
16-8
1/128-1/256
1/256-1/512
1/512-1/1024
1/1024-1/2048
0.008-0.004
0.004-0.0020
0.0020-0.0010
0.0010-0.0005
8-4
4-2
2-1
1-0.5
1/2048-1/4096
0.0005-0.0002
0.5-0.24
160-80
80-40
40-20
20-10
10-5
5-2.5
2.5-1.3
1.3-0.6
0.6-0.3
0.3-0.16
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
Class
Name
Very large boulders
Large boulders
Medium boulders
Small boulders
Large cobbles
2.5
5
blank
5
Small cobbles
Very coarse gravel
Coarse gravel
Medium gravel
Fine gravel
9
16
32
60
115
10
18
35
60
120
blank
blank
blank
blank
250
blank
blank
blank
blank
230
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
blank
0.16-0.08
blank
blank
blank
blank
blank
A.9
cubic
inch
liter
U.S.
gallon
cubic
foot
cubic
yard
cubic
meter
acre-foot
sec-footday
61.02
231
1,728
46,660
1
3.785
28.32
764.6
0.264 2
1
7.481
202
0.035 31
0.133 7
1
27
0.001 31
0.004 95
0.037 04
1
0.001
0.003 79
0.028 32
0.746 60
810.6 E-9
3.068 E-6
22.96 E-6
619.8 E-6
408.7 E-9
1.547 E-6
11.57 E-6
312.5 E-6
61,020
75.27 E+6
149.3 E+6
1,000
1,233,000
2,447,000
264.2
325,900
646,400
35.31
43,560
86,400
1.308
1,613
3,200
1
1,233
2,447
810.6 E-6
1
1.983
408.7 E-6
0.5042
1
gallon /
minute
liter /
second
acre-foot /
day
foot3 /
second
million gallon
/ day
meter3 /
second
1
15.85
226.3
0.063 09
1
14.28
0.004 419
0.070 05
1
0.002 228
0.035 31
0.504 2
0.001 440
0.022 82
0.325 9
63.09 E-06
0.001
0.014 28
448.8
694.4
28.32
43.81
1.983
3.068
1
1.547
0.646 3
1
0.028 32
0.043 82
15,850
1,000
70.04
35.31
22.82
A.10