ASCE HEC-RAS SeminarJanuary 25, 2006

Session 1BHydraulic 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

Flow Profiles Energy Losses

Selection of n-Values 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 + V2/2gFor a wide channel, V =q/d, and so,

H = d + q2/2gd2

For a given flow then,q2/2g = d2(H-d)

Specific Energy DiagramSpecific Energy q=5

0

1

2

3

4

5

0 1 2 3 4 5

H (ft)

d (

ft)

Specific Energy Diagram

Specific Energy Diagram

00.5

11.5

22.5

33.5

4

0 1 2 3 4 5 6 7

H (ft.)

De

pth

(ft

)

Energy 3cfsfEnergy 5cfsf

Critical Conditions

Critical flow occurs when the Froude Number

( ) 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.

dgV

Fr

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

0

1

2

3

4

5

0 1 2 3 4 5

H (ft)

d (

ft)

E1E2

ΔZ

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 from: g

Q

yA 2

h

c3

3

12

c gq

y

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 ChangeSpecific Energy Diagram

00.5

11.5

22.5

33.5

4

0 1 2 3 4 5 6 7

H (ft.)

De

pth

(ft

)

Energy 3cfsfEnergy 5cfsf

Bottom Change & Width Change

Specific Energy Diagram

00.5

11.5

22.5

33.5

4

0 1 2 3 4 5 6 7

H (ft.)

De

pth

(ft

)

Energy 3cfsfEnergy 5cfsf

Energy Analysis

The energy (for initial analysis) is assumed to be constant (no losses).

The energy equation provides:

222

22

121

21

22

1

1

Y2gY

qY

2gY

q

W

Qq channel wideof use theand

2g

V

2g

V22

YY

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?

1YY4.64

253

94.64

92

908.1Y

Hydraulic Jump

The only way flow can cross from super-critical 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:

1F8121

y

y 2

ri

i

s

is

3

1sL yy4

yy h

Specific Energy of Jump

Specific Energy q=5

0

1

2

3

4

5

0 1 2 3 4 5H (ft)

d (ft)

#REF!

ys

yi

Differential Equation of Channel Flow

By rearranging the Energy equation:

We get :

L2

2

221

2

11 h2gVP

Z2gVP Z

2

r

fo

F-1

SS

dxdy

3

c

3/10

n

oy

-1

yy

1

S dxdy

y

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 Non-Uniform 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

2

2/3

av

avavf

of

12

2

11

2

22fo

Lf

21o

L

2

222

2

111

R1.49

VnS

SSHH

x

2gV

y2gV

yxSS

xh

S xzz

S

h2gV

yZ2gV

yZ

Conditions Conditions

Rectangular 20 ft. wide channel with Rectangular 20 ft. wide channel with slope of 0.0005 and an n-value of 0.018 slope of 0.0005 and an n-value of 0.018 conveying 800 cfs, ends at an abrupt conveying 800 cfs, ends at an abrupt dropdrop

Yn=8.01

Yc=3.68Y(x)

4yc

.7yc

M2

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 n-values 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

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

543210 )( mnnnnnn

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

Slope 0.000485 n= 0.0178 Yn= 2.91 Estimated from Wide ChannelWidth (Ft) 18.6 Yc= 1.55

del y= 0.25Q(cfs)= 203Depth Area Hyd Radius Vel H dH Rav Vav Slope EGL dx X WSELEV Invert1.55 28.8 1.33 7.06 2.323 15 1.55 0.0072811.80 33.4 1.51 6.08 2.372 0.0495 1.417 6.57 0.00387 14.60 29.6 1.812149 0.0143682.05 38.1 1.68 5.34 2.490 0.1183 1.592 5.71 0.00250 58.73 88.3 2.090658 0.0428772.30 42.7 1.84 4.76 2.649 0.1589 1.760 5.05 0.00171 129.93 218.3 2.403728 0.1059482.55 47.4 2.00 4.29 2.834 0.1844 1.921 4.53 0.00122 250.80 469.1 2.775473 0.2276922.80 52.0 2.15 3.91 3.035 0.2012 2.075 4.10 0.00090 481.11 950.2 3.259015 0.4612352.91 54.1 2.22 3.76 3.130 0.0950 2.184 3.83 0.00074 377.19 1327.4 3.555192 0.644332.91 54.1 2.22 3.76 3.130 0.0000 2.217 3.76 0.00069 0.00 1327.4 3.555192 0.644332.91 54.1 2.22 3.76 3.130 0.0000 2.217 3.76 0.00069 0.00 1327.4 3.555192 0.644332.91 54.1 2.22 3.76 3.130 0.0000 2.217 3.76 0.00069 0.00 1327.4 3.555192 0.644332.91 54.1 2.22 3.76 3.130 0.0000 2.217 3.76 0.00069 0.00 1327.42.91 54.1 2.22 3.76 3.130 0.0000 2.217 3.76 0.00069 0.00 1327.42.91 54.1 2.22 3.76 3.130 0.0000 2.217 3.76 0.00069 0.00 1327.42.91 54.1 2.22 3.76 3.130 0.0000 2.217 3.76 0.00069 0.00 1327.42.91 54.1 2.22 3.76 3.130 0.0000 2.217 3.76 0.00069 0.00 1327.42.91 54.1 2.22 3.76 3.130 0.0000 2.217 3.76 0.00069 0.00 1327.42.91 54.1 2.22 3.76 3.130 0.0000 2.217 3.76 0.00069 0.00 1327.42.91 54.1 2.22 3.76 3.130 0.0000 2.217 3.76 0.00069 0.00 1327.42.91 54.1 2.22 3.76 3.130 0.0000 2.217 3.76 0.00069 0.00 1327.42.91 54.1 2.22 3.76 3.130 0.0000 2.217 3.76 0.00069 0.00 1327.42.91 54.1 2.22 3.76 3.130 0.0000 2.217 3.76 0.00069 0.00 1327.42.91 54.1 2.22 3.76 3.130 0.0000 2.217 3.76 0.00069 0.00 1327.42.91 54.1 2.22 3.76 3.130 0.0000 2.217 3.76 0.00069 0.00 1327.42.91 54.1 2.22 3.76 3.130 0.0000 2.217 3.76 0.00069 0.00 1327.42.91 54.1 2.22 3.76 3.130 0.0000 2.217 3.76 0.00069 0.00 1327.42.91 54.1 2.22 3.76 3.130 0.0000 2.217 3.76 0.00069 0.00 1327.42.91 54.1 2.22 3.76 3.130 0.0000 2.217 3.76 0.00069 0.00 1327.42.91 54.1327 2.22 3.76 3.129953 0 2.217 3.76 0.00069 0.00 1327.4

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0 500 1000 1500 2000

Distance from Overfall

Wat

er e

leva

tio

n

Water Surface

Channel Invert

For Channels that Are NOT Regular – Standard Step.

For channels that are irregular, the cross-sections 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

Line GradeEnergy of Slope iveRepresenatS

LengthReach WeightedFlowL

LossStructureH

2g

α1V

2g

VαCSLH

HH2g

α1VWS

2g

VαWS

f

s

21

222

cfB

sB

21

1

222

2

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. Gradually Varied because hydrostatic Pressure is assumed

2. One-Dimensional3. Steady because no time term is

present4. 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