ASCE HEC-RAS Seminar
January 25, 2006
Session 1B
Hydraulic Data and Fundamental
Behavior Affected by Uncertainty
Quote
Even if there is only one possible unified theory, it
is just a set of rules and equations. What is it
that breathes fire into the equations and makes
a universe for them to describe?
The usual approach of science of constructing a
mathematical model cannot answer the
questions of why there should be a universe to
describe. Why does the universe go to all the
bother of existing?”

Stephen Hawking
Topics of Session

Review of Steady, Non-Uniform Flow



Selection of n-Values




Flow Profiles
Energy Losses
Understanding Variation
Effects of Uncertainty in Loss Calcs
Uncertainty in Section Geometry
Importance of Thresholds of Behavior
Uniform (Normal) Conditions vs
Non-Uniform Conditions
In “normal” flow, the water surface is parallel to
the bed slope and the EGL. This is not a
normal occurrence. All flow profiles only
approach normal depth asymptotically, and it
can take a great distance for the depth to
equal normal.
In non-uniform flow the depth changes so the
water surface changes; we need to predict
the change.
Specific Energy of Flow
Specific Energy, H is the flow energy
measured W.R.T. the channel bottom:
H = d + V2/2g
For a wide channel, V =q/d, and so,
H = d + q2/2gd2
For a given flow then,
q2/2g = d2(H-d)
Specific Energy Diagram
Specific Energy q=5
5
4
d (ft)
3
2
1
0
0
1
2
3
H (ft)
4
5
Specific Energy Diagram
Depth (ft)
Specific Energy Diagram
4
3.5
3
2.5
2
1.5
1
0.5
0
Energy 3cfsf
Energy 5cfsf
0
1
2
3
4
H (ft.)
5
6
7
Critical Conditions
Critical flow occurs when the Froude Number
(
Fr 
V
gd
)
is exactly 1.
This is the point the flow can have min. energy,
and depends only on flow rate not on
geometry or roughness.
If flow depth is greater than critical depth it is
sub-critical, if less it’s super-critical.
Direction of Information
Transfer




If flow is sub-critical (Fr < 1) the flow depth is
affected upstream.
If the flow is super-critical (Fr > 1) the flow
depth is affected downstream.
Toss a rock in the flow, if ripples move
upstream against the flow it is sub-critical.
The Flow Regime also affects the changes in
depth caused by channel transitions.
Channel Transitions




In tranquil flow, a bottom change up causes a
depth reduction, a width decrease also
causes a depth reduction.
In rapid flow, a bottom change down causes
a small depth decrease and a width increase
also causes a depth decease.
THIS BEHAVIOR IS COUNTER INTUITIVE!
Make a quick sketch to see if the behavior is
possible
Specific Energy Diagram
Specific Energy q=5
5
ΔZ
4
d (ft)
3
2
1
0
0
1
2
3
H (ft) E2
4
E1
5
For Channels That are NOT Wide
In any case where the width is less than
about 10 times the depth, the use of
the depth as the hydraulic radius is less
accurate. In those cases use:


V=Q/A
Depth = A/Top Width ( the hydraulic
depth)
Thus the critical depth is determined
A
Q
q 
from:

y  
3
2
2
c
yh
g
c
g
1
3
Transitions—Specific Energy Analysis




Calculate q, Fr and E
Determine if Rapid or Tranquil
Determine if Energy is increased or
decreased
Sketch Specific Energy Diagram
Transitions (Analysis)



In general, transitions can be changes
in width or changes in bottom elevation.
The basis of the water surface response
to a transition is the specific energy
diagram.
For a more advanced analysis, energy
losses must be incorporated.
Width Change
Depth (ft)
Specific Energy Diagram
4
3.5
3
2.5
2
1.5
1
0.5
0
Energy 3cfsf
Energy 5cfsf
0
1
2
3
4
H (ft.)
5
6
7
Bottom Change & Width Change
Depth (ft)
Specific Energy Diagram
4
3.5
3
2.5
2
1.5
1
0.5
0
Energy 3cfsf
Energy 5cfsf
0
1
2
3
4
H (ft.)
5
6
7
Energy Analysis
The energy (for initial analysis) is
assumed to be constant (no losses).
The energy equation provides:
2
2
V1
V2
 Y1 
 Y2
2g
2g
Q
and the use of wide channel q 
W
2
2
q1
q2
 Y1 
 Y2
2
2
2gY1
2gY2
Solution in Simple Case
The equation allows calculation of depths,
widths, velocities, energies or flow,
depending on what is given. The
definition of continuity, energy and
elevation is often combined with the
energy equation to get a solution.
Simple Example
If the flow is 30 cfs in a 10 ft wide
channel with a depth of 3 ft and the
width changes to 6 ft at the same time
as the bottom is raised one ft, what is
the depth and change in WSEL?
9
25
3
 Y 1
2
64.4  9
64.4  Y
Y  1.908
Hydraulic Jump
The only way flow can cross from supercritical to sub-critical regimes is through
a hydraulic jump.
The location of a jump is determined be
the relationship of the sequent depth to
the incoming flow depth.
Hydraulic jump


The energy lost in a jump is large!
Sequent depth is:
ys 1

yi 2


1  8F  1
2
ri
y s  y1 
3
hL 
4  ys  yi
Specific Energy of Jump
Specific Energy q=5
5
4
ys
d (ft)
3
#REF!
yi
2
1
0
0
1
2
H (ft)
3
4
5
Differential Equation of
Channel Flow
By rearranging the Energy equation:
2
V1
P2 V22

 Z1  
 Z2  hL
 2g
 2g
P1
We get :
dy S o  Sf

dx 1 - Fr2
  yn 
1   
y
dy


 So
3
dx

 yc 
 1-  y 
 

10 / 3






The Gradually Varied Flow
Profiles
The profile depends on:



The ratio of flow depth to normal depth
The ratio of flow depth to critical depth
The bed slope

Sustaining




Mild
Steep
Critical
Non-Sustaining


Adverse
Horizontal
Flow Profiles






Draw critical and normal depth on
channel profile, number zones from
outer zone
Mild yn > yc
M1, M2, M3
Steep yn< yc
S1, S2, S3
Critical yn = yc
C1, C2, C3
Horizontal no normal
H2, H3
Adverse no normal
A2, A3
The Differential Equation of NonUniform Flow solved by steps:
For channels with regular geometry the
profile is calculated by balancing the
energy equation for an assumed water
depth. Resulting is a calculated distance
along the channel to the point where
the assumed depth will occur. This is
“Direct Step”.
Equations
V12
V22
Z1  y 1 
 Z2  y2 
 hL
2g
2g
z1  z 2
hL
So 
Sf 
x
x
V22  
V12 

So  Sf   x   y 2     y1  
2g  
2g 

H 2  H1
x 
Sf  So
 n av  Vav 
Sf  

2/3
 1.49  R av 
2
Conditions
Rectangular 20 ft. wide channel with slope of 0.0005
and an n-value of 0.018 conveying 800 cfs, ends at
an abrupt drop
Yn=8.01
Yc=3.68
M2
Y(x)
.7yc
4yc
Calculations
y
A
R
V
H
dH Rav Vav Sf*k dX
-
-
-
X
3.68 73.6 2.68 10.9 5.52 -
-
15
4.68 93.6 3.19 8.6
5.82 .3
2.94 9.71 3.27 108
123
5.68 113. 3.62 7.0
6.45 .63 3.41 7.79 1.72 518
641
6.68 133 4.00 6.0
7.24 .79 3.81 6.52 1.04 1449 2090
7.68 153 4.34 5.21 8.10 .86 4.17 5.60 6.82 4750 6840
7.93 158 4.42 5.0
8.32 .23 4.38 5.13 5.36 6212 13052
Numerical Sensitivity?



What id the steps in Water depth were
0.5 ft?
0.01 ft?
How close is close enough?
Roughness Estimates


For lined channels the theoretical
description of flow behavior is useful.
Manning’s n-values, f, C and CH can be
used.
There is little advantage to not using nvalues but f and C are more
fundamentally correct
Roughness


It it essential to recognize that open
channel flow has variable flow geometry
rather than only variable velocity (as in
a pipe). Thus, the relative roughness
(ks/D) changes.
As the roughness changes so does the
n-value (f and C also).
Roughness






Roughness has components that are
considered separately: 2-28 River
Engineering
Channel material
Vegetation
Alignment
Channel Irregularity
Channel Variation
Roughness Factors
n  (n0  n1  n2  n3  n4 )m5
where:

no = Base value for straight uniform channels

n1 = Additive value due to cross-section irregularity

n2 = Additive value due to variations of the channel

n3 = Additive value due to obstructions
 n4 = Additive value due to vegetation

m5 = Mulitiplication factor due to sinuosity
Channel Roughness Catalogs




“Rules of Thumb ”
Textural catalogs
Photographic Catalogs of measured
roughness values USGS Barnes
REMEMBER the boundary roughness
can change from bed-form changes
induced by the flow.
Barnes
Barnes 2
Barnes 3
Dealing With Roughness
Uncertainty




Some Engineers have told me “n-value you pick
doesn’t matter, nobody knows the correct
number”… WTF, over??????
I use a range of reasonable values to calculate
how the variables you are examining change
with roughness changes.
Does your decision change??
What is Sensitivity of your situation to the
uncertainty?
Compound Channels


In most real channels the change of
channel area as the depth changes is
not a smooth function. There are
frequently floodplains where width
increases enormously with a small
change in depth.
These situations are called compound
channels.
Channel Geometry
Analysis
Analysis




Sub-sections of the channel are
identified with a zone of equal n-value.
Water-to-water shear is neglected.
Energy slope is the same for all zones.
Sum of sub-section discharge is total
discharge.
Alternative Analysis



Would “average” n-value over entire
channel be acceptable? When ? Why?
What is influence of neglecting the
water-to-water shear??
What is a practical limit to sub-section
division?
Uncertainty in Simple Case
0.0178
Yn=
Yc=
2.91
1.55
Estimated from Wide Channel
Vel
7.06
6.08
5.34
4.76
4.29
3.91
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
H
2.323
2.372
2.490
2.649
2.834
3.035
3.130
3.130
3.130
3.130
3.130
3.130
3.130
3.130
3.130
3.130
3.130
3.130
3.130
3.130
3.130
3.130
3.130
3.130
3.130
3.130
3.130
3.129953
dH
Rav
Vav
Slope EGL
dx
0.0495
0.1183
0.1589
0.1844
0.2012
0.0950
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0
1.417
1.592
1.760
1.921
2.075
2.184
2.217
2.217
2.217
2.217
2.217
2.217
2.217
2.217
2.217
2.217
2.217
2.217
2.217
2.217
2.217
2.217
2.217
2.217
2.217
2.217
2.217
6.57
5.71
5.05
4.53
4.10
3.83
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
3.76
0.00387
0.00250
0.00171
0.00122
0.00090
0.00074
0.00069
0.00069
0.00069
0.00069
0.00069
0.00069
0.00069
0.00069
0.00069
0.00069
0.00069
0.00069
0.00069
0.00069
0.00069
0.00069
0.00069
0.00069
0.00069
0.00069
0.00069
14.60
58.73
129.93
250.80
481.11
377.19
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
X
15
29.6
88.3
218.3
469.1
950.2
1327.4
1327.4
1327.4
1327.4
1327.4
1327.4
1327.4
1327.4
1327.4
1327.4
1327.4
1327.4
1327.4
1327.4
1327.4
1327.4
1327.4
1327.4
1327.4
1327.4
1327.4
1327.4
WSELEV
1.55
1.812149
2.090658
2.403728
2.775473
3.259015
3.555192
3.555192
3.555192
3.555192
Invert
0.007281
0.014368
0.042877
0.105948
0.227692
0.461235
0.64433
0.64433
0.64433
0.64433
4.00
3.50
Water elevation
Slope
0.000485
n=
Width (Ft)
18.6
del y=
0.25
Q(cfs)=
203
Depth
Area
Hyd Radius
1.55
28.8
1.33
1.80
33.4
1.51
2.05
38.1
1.68
2.30
42.7
1.84
2.55
47.4
2.00
2.80
52.0
2.15
2.91
54.1
2.22
2.91
54.1
2.22
2.91
54.1
2.22
2.91
54.1
2.22
2.91
54.1
2.22
2.91
54.1
2.22
2.91
54.1
2.22
2.91
54.1
2.22
2.91
54.1
2.22
2.91
54.1
2.22
2.91
54.1
2.22
2.91
54.1
2.22
2.91
54.1
2.22
2.91
54.1
2.22
2.91
54.1
2.22
2.91
54.1
2.22
2.91
54.1
2.22
2.91
54.1
2.22
2.91
54.1
2.22
2.91
54.1
2.22
2.91
54.1
2.22
2.91
54.1327
2.22
3.00
2.50
Water Surface
2.00
Channel Invert
1.50
1.00
0.50
0.00
0
500
1000
1500
Distance from Overfall
2000
For Channels that Are NOT
Regular – Standard Step.
For channels that are irregular, the crosssections are located at given positions.
Therefore, a guess is made of the water
level at the next section. Based on that
guess the energy loss is calculated, the
calculated water level is then compared
to the guess, and the guess updated
until an acceptable ‘closure’ at that
section is obtained.
Standard Step Data
Standard Step Calculations


All irregular channels require the use of
standard step. Because the calculations
of the energy loss is tedious the method
is best computerized.
There are many issues to consider in
the calculation scheme.


TOTAL Head Loss
Average Roughness in Each location and
between sections
Standard Step Equations
α 2 V22
α1V12
WS2 
 WS1 
 H B  Hs
2g
2g
α 2 V22 α1V12
H B  LSf  C c

2g
2g
H s  Structure
Loss
L  Flow Weighted Reach Length
Sf  Represenat ive Slope of Energy Grade Line
Standard Step Calculation
Procedure
Beginning at known conditions, guess Y2,
with channel shape calculate V2, then
S2, and solve for what Y2 satisfies the
original energy equation. If guess and
calculated value are the “same”, that is
“correct” answer. Otherwise guess
again.
The Limitations of Standard
Step Method
1.
2.
3.
4.
Gradually Varied because hydrostatic
Pressure is assumed
One-Dimensional
Steady because no time term is
present
Small channel Slope (10%-20%)
because y and H are assumed
collinear.
Homework

Look at the following publications in the
references CD:





The origin and Derivation of Ia/S in the
Runoff Curve Number System
NEH Part 630 Hydrology and HEH 4 (old)
HEC-HMS Documentation
Basic Hydraulic Principles
River Engineering