Energy or Energy Head

1
Energy or Energy Head
Energy Head -Elevation Head -Velocity Head
-Total Head Momentum Open Channel

• Elevation head
• Velocity head
• Total head

2
Energy or Energy Head

• The total energy of water moving through a
  channel is expressed in total head in feet of
  water.
• This is simply the sum of the the elevation above
  a datum (elevation head), the pressure head and
  the velocity head.
• The elevation head is the vertical distance from
  a datum to a point in the stream.
• The velocity head is expressed by

Energy Head -Elevation Head -Velocity Head
-Total Head Momentum Open Channel
3
Energy Head
Graphical depiction of elevation head, velocity
head, and total head. Total head is the sum of
velocity head, depth and elevation head.
Energy Head -Elevation Head -Velocity Head
-Total Head Momentum Open Channel
Energy Grade Line
headloss
Hydraulic Grade Line
Veloctiy head
(water surface)
Depth1
Channel Bottom
Elevation Head
Depth2
Datum
4
Momentum Equation
Energy Head Momentum -Equation -Forces Open
Channel
Hydrostatic Forces Friction Forces
Weight External Forces
5
Hydrostatic Forces

• Hydrostatic Forces are the forces placed on a
  control volume by the surrounding water.
• The strength of the force is based on depth and
  can be seen in the following relationship

Energy Head Momentum -Equation -Forces Open
Channel
Hydrostatic Forces
Control Volume
Hydrostatic Forces Friction Forces
Weight External Forces
6
Friction Forces
Energy Head Momentum -Equation -Forces Open
Channel
The friction force on a control volume is due to
the water passing the channel bottom and depends
on the roughness of the channel.
Control Volume

Friction Force
Hydrostatic Forces Friction Forces
Weight External Forces
7
Weight
The weight of a control volume is due to the
gravitational pull on the its mass.
Energy Head Momentum -Equation -Forces Open
Channel
Weight mg
Control Volume
Weight
Hydrostatic Forces Friction Forces
Weight External Forces
8
External Forces
External Forces (Fd) the forces created by a
control volume striking a stationary object.
External Forces can be explained by the following
equation
Energy Head Momentum -Equation -Forces Open
Channel
Fd1/2CdrAv2
Hydrostatic Forces Friction Forces
Weight External Forces
9
Steady vs. Unsteady Flow

• Fluid properties including velocity, pressure,
  temperature, density, and viscosity vary in time
  and space.
• A fluid it termed steady if the depth of flow
  does not change or can be assumed constant during
  a specific time interval.
• Flow is considered unsteady if the depth changes
  with time.

Energy Head Momentum Open Channel -Steady -vs-
Unsteady -Uniform -vs- Nonuniform -Supercitical
-vs- subcritical -Equations
10
Uniform and Nonuniform Flow

• Uniform Flow is an equilibrium flow such that the
  slope of the total energy equals the bottom
  slope.
• Nonuniform Flow is a flow of water through a
  channel that gradually changes with distance.

Energy Head Momentum Open Channel -Steady -vs-
Unsteady -Uniform -vs- Nonuniform -Supercitical
-vs- subcritical -Equations
11
Super -vs.- Sub Critical
Energy Head Momentum Open Channel
-Steady-vs.-Unsteady -Uniform-vs. Nonuniform
-Sub/Supercritical -Equations
12
Critical flow a demonstration
If a stone is dropped into a body of water, with
no velocity, the waves formed by the water are
fairly circular. This is similar to sub-critical
flow.
Energy Head Momentum Open Channel
-Steady-vs.-Unsteady -Uniform-vs. Nonuniform
-Sub/Supercritical -Equations
No velocity
13
Critical flow a demonstration
Now, if a velocity is added to the body of water,
the waves become unsymmetrical, increasing to the
downstream side. This happens as the velocity
approaches critical flow. Notice that the wave
still moves upstream, though slower than the
downstream wave.
Energy Head Momentum Open Channel
-Steady-vs.-Unsteady -Uniform-vs. Nonuniform
-Sub/Supercritical -Equations
Small velocity
14
Critical flow a demonstration
Energy Head Momentum Open Channel
-Steady-vs.-Unsteady -Uniform-vs. Nonuniform
-Sub/Supercritical -Equations
Now if a large velocity is added to the body of
water, the wave patterns only go in one
direction. This represents the point when flow
has gone beyond critical, into the supercritical
region.
Large velocity
15
Froude number
The Froude number is a numerical value that
describes the type of flow present (critical,
supercritical, subcritical), and is represented
by the following equation for a rectangular
channel
Energy Head Momentum Open Channel
-Steady-vs.-Unsteady -Uniform-vs. Nonuniform
-Sub/Supercritical -Froude number -Equations
NF Froude number v mean velocity of flow g
acceleration of gravity dm mean (hydraulic)
depth
16
Froude number
The generalized formula for the Froude number is
as follows
Energy Head Momentum Open Channel
-Steady-vs.-Unsteady -Uniform-vs. Nonuniform
-Sub/Supercritical -Froude number -Equations
Fr Froude number Q Flow rate in the channel B
Top width of water surface A Area of the
channel
17
Froude number - mean depth
Energy Head Momentum Open Channel
-Steady-vs.-Unsteady -Uniform-vs. Nonuniform
-Sub/Supercritical -Froude number -Equations

• Mean depth is a ratio of the width of the free
  water surface to the cross-sectional area of the
  channel.

18
Froude number

• The Froude number can then be used to quantify
  the type of flow.
• If the Froude number is less than 1.0, the flow
  is subcritical. The flow would would be
  characterized as tranquil.
• If the Froude number is equal to 1.0, the flow is
  critical.
• If the Froude number is greater than 1.0, the
  flow is supercritical and would be characterized
  as rapid flowing. This type of flow has a high
  velocity which can be potentially damaging.

Energy Head Momentum Open Channel
-Steady-vs.-Unsteady -Uniform-vs. Nonuniform
-Sub/Supercritical -Froude number -Equations
19
Super-vs.-Subcritical

• Critical depth can also be determined by
  constructing a Specific Energy Curve.
• The critical depth is the point on the curve with
  the lowest specific energy.
• Any depth greater than critical depth is
  subcritical flow and any depth less than is
  supercritical flow.

Energy Head Momentum Open Channel
-Steady-vs.-Unsteady -Uniform-vs. Nonuniform
-Sub/Supercritical -Equations
20
Super-vs.-Subcritical
21
Open Channel Equations
Energy Head Momentum Open Channel -Steady -vs-
Unsteady -Uniform -vs- Nonuniform -Supercitical
-vs- subcritical -Equations Chezy Manning Bernou
lli St. Venant

• Chezy Equation
• Mannings Equation
• Bernoulli Equation
• St. Venant Equations

22
Chezy Equation
Energy Head Momentum Open Channel -Chezy
Equation -Mannings -Bernoulli -St. Venant

• In 1769, the French engineer Antoine Chezy
  developed the first uniform-flow formula.

• The formula was derived based on two assumptions.
  First, Chezy assumed that the force resisting
  the flow per unit area of the stream bed is
  proportional to the square of the velocity (KV2),
  with K being a proportionality constant.

23
Chezy Equation
Energy Head Momentum Open Channel -Chezy
Equation -Mannings -Bernoulli -St. Venant

• The second assumption was that the channel was
  undergoing uniform flow.
• The difficulty with this formula is determining
  the value of C, which is the Chezy resistance
  factor. There are three different formulas for
  determining C, the G.K. Formula, the Bazin
  Formula, and the Powell Formula.

24
Chezy Equation
Energy Head Momentum Open Channel -Chezy
Equation -Mannings -Bernoulli -St. Venant

• Later on, when Manning's equation was developed
  in 1889, a relationship between Mannings n and
  Chezys C was established.
• Finally in 1933, the Manning equation was
  suggested for international use rather than
  Chezys Equation.

25
Mannings Equation
Energy Head Momentum Open Channel -Chezy
Equation -Mannings -Bernoulli -St. Venant

• In 1889 Robert Manning, an Irish engineer,
  presented the following formula to solve open
  channel flow.

V mean velocity in fps R hydraulic radius in
feet S the slope of the energy line n
coefficient of roughness
The hydraulic radius (R) is a ratio of the water
area to the wetted perimeter.
26
Mannings Equation
Energy Head Momentum Open Channel -Chezy
Equation -Mannings -Bernoulli -St. Venant

• This formula was later adapted to obtain a flow
  measurement. This is done by multiplying both
  sides by the area.

• Mannings equation is the most widely used of all
  uniform-flow formulas for open channel flow,
  because of its simplicity and satisfactory
  results it produces in real-world applications.

27
Mannings Equation
Energy Head Momentum Open Channel -Chezy
Equation -Mannings -Bernoulli -St. Venant

• Note that the equation expressed in the previous
  slide was the English version of Mannings
  equation.
• There is also a metric version of Mannings
  equation, which replaces the 1.49 with 1. This
  is done because of unit conversions.
• The metric equation is

28
Bernoulli Equation
Energy Head Momentum Open Channel -Chezy
Equation -Mannings -Bernoulli -St. Venant

• The Bernoulli equation is developed from the
  following equation

This equation states that the elevation (z) plus
the depth (y) plus the velocity head (V12/2g) is
a constant. The difference being the headlosses
- hL
29
Bernoulli Equation

• This equation was then adapted by making a few
  assumptions.
• First, the head loss due to friction is equal to
  zero. This means the channel is perfectly
  frictionless surface.
• Second, that alpha1 is equal to alpha2 which is
  equal to 1. The alphas are in the original
  equation to account for a non-uniform velocity
  distribution. In this case we will assume a
  uniform distribution which produces the following
  equation

Energy Head Momentum Open Channel -Chezy
Equation -Mannings -Bernoulli -St. Venant
30
Bernoulli Equation
Energy Head Momentum Open Channel -Chezy
Equation -Mannings -Bernoulli -St. Venant
A simplified version of the formula is given
below
31
Bernoulli Equation
Energy Head Momentum Open Channel -Chezy
Equation -Mannings -Bernoulli -St. Venant

• Some comments on the Bernoulli equation
• Energy only
• Headloss in terms of energy
• Cannot calculate forces
• Limited Effect in rapidly varying flow

32
St. Venant Equations
Energy Head Momentum Open Channel -Chezy
Equation -Mannings -Bernoulli -St. Venant
The two equations used in modeling are the
continuity equation and the momentum equation.
Continuity equation
Momentum Equation
33
St. Venant Equations
The Momentum Equation can often be simplified
based on the conditions of the model.
Energy Head Momentum Open Channel -Chezy
Equation -Mannings -Bernoulli -St. Venant
Unsteady -Nonuniform
Steady - Nonuniform
Diffusion or noninertial
Kinematic
34
Simulating the Hydrologic Response
Model Types Precipitation Losses Modeling
Losses Model Components
35
Model Types
Model Types Precipitation Losses Modeling
Losses Model Components

• Empirical
• Lumped
• Distributed

36
Precipitation

• magnitude, intensity, location, patterns, and
  future estimates of the precipitation.
• In lumped models, the precipitation is input in
  the form of average values over the basin. These
  average values are often referred to as mean
  aerial precipitation (MAP) values.
• MAP's are estimated either from 1)
  precipitation gage data or 2) NEXRAD
  precipitation fields.

Model Types Precipitation -Thiessen
-Isohyetal -Nexrad Losses Modeling Losses Model
Components
37
Precipitation (cont.)

• If precipitation gage data is used, then the
  MAP's are usually calculated by a weighting
  scheme.
• a gage (or set of gages) has influence over an
  area and the amount of rain having been recorded
  at a particular gage (or set of gages) is
  assigned to an area.
• Thiessen method and the isohyetal method are
  two of the more popular methods.

Model Types Precipitation -Thiessen
-Isohyetal -Nexrad Losses Modeling Losses Model
Components
38
Thiessen
Model Types Precipitation -Thiessen
-Isohyetal -Nexrad Losses Modeling Losses Model
Components

• Thiessen method is a method for areally weighting
  rainfall through graphical means.

39
Isohyetal
Model Types Precipitation -Thiessen
-Isohyetal -Nexrad Losses Modeling Losses Model
Components

• Isohyetal method is a method for areally
  weighting rainfall using contours of equal
  rainfall (isohyets).

40
NEXRAD
Model Types Precipitation -Thiessen
-Isohyetal -Nexrad Losses Modeling Losses Model
Components

• Nexrad is a method of areally weighting rainfall
  using satellite imaging of the intensity of the
  rain during a storm.

41
Losses

• modeled in order to account for the destiny of
  the precipitation that falls and the potential of
  the precipitation to affect the hydrograph.
• losses include interception, evapotranspiration,
  depression storage, and infiltration.
• Interception is that precipitation that is
  caught by the vegetative canopy and does not
  reach the ground for eventual infiltration or
  runoff.
• Evapotranspiration is a combination of
  evaporation and transpiration and was previously
  discussed.
• Depression storage is that precipitation that
  reaches the ground, yet, as the name suggests, is
  stored in small surface depressions and is
  generally satisfied during the early portion of a
  storm event.

Model Types Precipitation Losses Modeling
Losses Model Components
42
Modeling Losses

• simplistic methods such as a constant loss
  method may be used.
• A constant loss approach assumes that the soil
  can constantly infiltrate the same amount of
  precipitation throughout the storm event. The
  obvious weaknesses are the neglecting of spatial
  variability, temporal variability, and recovery
  potential.
• Other methods include exponential decays (the
  infiltration rate decays exponentially),
  empirical methods, and physically based methods.
• There are also combinations of these methods.
  For example, empirical coefficients may be
  combined with a more physically based equation.
  (SAC-SMA for example)

Model Types Precipitation Losses Modeling Losses
-SAC-SMA Model Components
43
Simulating Watershed ResponseInfiltration
Long Term vs.- Short Infiltration Evapotranspirat
ion Unit Hydrograph Timing Routing
Infiltration or losses - this section describes
the action of the precipitation infiltrating into
the ground. It also covers the concept of
initial abstraction, as it is generally
considered necessary to satisfy the initial
abstraction before the infiltration process
begins.
44
Simulating Watershed ResponseInfiltration
Long Term vs.- Short Infiltration Evapotranspirat
ion Unit Hydrograph Timing Routing
Initial Abstraction - It is generally assumed
that the initial abstractions must be satisfied
before any direct storm runoff may begin. The
initial abstraction is often thought of as a
lumped sum (depth). Viessman (1968) found that
0.1 inches was reasonable for small urban
watersheds. Would forested rural watersheds be
more or less?
45
Simulating Watershed ResponseInfiltration
Long Term vs.- Short Infiltration Evapotranspirat
ion Unit Hydrograph Timing Routing
Forested rural watersheds would probably have a
higher initial abstraction. The Soil Conservation
Service (SCS) now the NRCS uses a percentage of
the ultimate infiltration holding capacity of the
soil - i.e. 20 of the maximum soil retention
capacity.
46
Simulating Watershed ResponseInfiltration
Long Term vs.- Short Infiltration Evapotranspirat
ion Unit Hydrograph Timing Routing
Infiltration is a natural process that we attempt
to mimic using mathematical processes. Some of
the mathematical process or simulation methods
are conceptual while others are more physically
based.
47
Simulating Watershed ResponseInfiltration
Constant Infiltration Rate A constant
infiltration rate is the most simple of the
methods. It is often referred to as a phi-index
or f-index. In some modeling situations it is
used in a conservative mode. The saturated soil
conductivity may be used for the infiltration
rate. The obvious weakness is the inability to
model changes in infiltration rate. The phi-index
may also be estimated from individual storm
events by looking at the runoff hydrograph.
Long Term vs.- Short Infiltration Evapotranspirat
ion Unit Hydrograph Timing Routing
48
Simulating Watershed ResponseInfiltration
Long Term vs.- Short Infiltration Evapotranspirat
ion Unit Hydrograph Timing Routing
Constant Percentage Method Another very
simplistic approach - this method assumes that
the watershed is capable of infiltrating or
using a value that is proportional to rainfall
intensity. The constant percentage rate can be
calibrated for a basin by again considering
several storms and calculating the percentage by
49
Constant Percentage Example
Long Term vs.- Short Infiltration Evapotranspirat
ion Unit Hydrograph Timing Routing
2
77.5 infiltrates
1
0
50
Simulating Watershed ResponseInfiltration
Long Term vs.- Short Infiltration Evapotranspirat
ion Unit Hydrograph Timing Routing
Exponential Decay This is purely a mathematical
function - of the following form
51
Simulating Watershed ResponseInfiltration
Long Term vs.- Short Infiltration Evapotranspirat
ion Unit Hydrograph Timing Routing
Exponential Decay
Effect of fo or fc
52
Simulating Watershed ResponseInfiltration
Long Term vs.- Short Infiltration Evapotranspirat
ion Unit Hydrograph Timing Routing
Exponential Decay
Effect of K
53
Simulating Watershed ResponseInfiltration
Long Term vs.- Short Infiltration Evapotranspirat
ion Unit Hydrograph Timing Routing
SCS Curve Number
Soil Conservation Service is an empirical method
of estimating EXCESS PRECIPITATION We can imply
that P - Pe F
54
SCS (NRCS) Runoff Curve Number

• The basic relationships used to develop the curve
  number runoff prediction technique are described
  here as background for subsequent discussion.
  The technique originates with the assumption that
  the following relationship describes the water
  balance of a storm event.

where F is the actual retention on the watershed,
Q is the actual direct storm runoff, S is the
potential maximum retention, and P is the
potential maximum runoff
55
More Modifications

• At this point in the development, SCS redefines S
  to be the potential maximum retention
• SCS defines Ia in terms of S as Ia 0.2S
• and since the retention, F, equals effective
  precipitation minus runoff F (P-Ia) - Q
• Substituting gives the familiar SCS
  rainfall-runoff

56
Estimating S

• The difficult part of applying this method to a
  watershed is the estimation of the watersheds
  potential maximum retention, S.
• SCS developed the concept of the dimensionless
  curve number, CN, to aid in the estimation of S.
• CN is related to S as follows

CN ranges from 1 to 100 (not really!)
57
Determine CN

• The Soil Conservation Service has classified over
  8,500 soil series into four hydrologic groups
  according to their infiltration characteristics,
  and the proper group is determined for the soil
  series found.
• The hydrologic groups have been designated as A,
  B, C, and D.
• Group A is composed of soils considered to have a
  low runoff potential. These soils have a high
  infiltration rate even when thoroughly wetted.
• Group B soils have a moderate infiltration rate
  when thoroughly wetted,
• while group C soils are those which have slow
  infiltration rates when thoroughly wetted.
• Group D soils are those which are considered to
  have a high potential for runoff, since they have
  very slow infiltration rates when thoroughly
  wetted (SCS, 1972).

58
Adjust CNs
59
SAC-SMA
Model Types Precipitation Losses Modeling Losses
-SAC-SMA Model Components

• The Sacramento Soil Moisture Accounting Model
  (SAC-SMA) is a conceptual model of soil moisture
  accounting that uses empiricism and lumped
  coefficients to attempt to mimic the physical
  constraints of water movement in a natural system.

60
Runoff

• Runoff is essentially the excess precipitation
  - the precipitation minus the losses.
• In the NWSRFS, runoff is modeled through the
  use of the SAC-SMA or an antecedent precipitation
  index (API) model.
• Runoff is transformed to streamflow at the
  basin outlet via a unit hydrograph.
• In actuality, all forms of surface and
  subsurface flow that reach a stream channel and
  eventually the outlet are modeled through the use
  of the unit hydrograph.

Model Types Precipitation Losses Modeling
Losses Model Components -Runoff -Unit
Hydrograph
61
Unit Hydrograph

• The hydrograph that results from 1-inch of excess
  precipitation (or runoff) spread uniformly in
  space and time over a watershed for a given
  duration.
• The key points
• 1-inch of EXCESS precipitation
• Spread uniformly over space - evenly over the
  watershed
• Uniformly in time - the excess rate is constant
  over the time interval
• There is a given duration

Model Types Precipitation Losses Modeling
Losses Model Components -Runoff -Unit
Hydrograph
62
Linearity of Unit Hydrograph

• In addition, when unit hydrograph theory is
  applied, it is assumed that the watershed
  responds linearly.
• Meaning that peak flow from 2 inches of excess
  will be twice that of 1 inch of excess

63
Derived Unit Hydrograph
64
Derived Unit Hydrograph
65
Derived Unit Hydrograph

• Rules of Thumb
• the storm should be fairly uniform in nature
  and the excess precipitation should be equally as
  uniform throughout the basin. This may require
  the initial conditions throughout the basin to be
  spatially similar.
• Second, the storm should be relatively constant
  in time, meaning that there should be no breaks
  or periods of no precipitation.
• Finally, the storm should produce at least an
  inch of excess precipitation (the area under the
  hydrograph after correcting for baseflow).

66
Synthetic Unit Hydrograph

• SCS
• Snyder
• Clark - (time-area)

67
SCS - Dimensionless UHG
68
SCS - Dimensionless UHG
69
SCS - Dimensionless UHG
70
Time-Area
71
Time-Area
72
Time-Area
73
Stream Routing

• ... stream routing is used to account for storage
  and translation effects as a runoff hydrograph
  travels from the outlet of one basin through the
  next downstream basin.
• Most of the time, channels act as reservoirs
  and have the effect of attenuating the
  hydrograph.
• 2 basic types of flow or channel routing
• hydrologic
• hydraulic

74
Typical Effect of Routing
75
Lakes, Reservoirs, Impoundments,

• ...have the effect of storing flow and
  attenuating hydrographs.
• Reservoirs (and impoundments) are modeled with
  some form of routing.
• hydrologic and hydraulic routing may be
  applicable although most often, hydrologic
  routing is used in reservoir routing for normal
  flow conditions.
• During failure scenarios an unsteady flow model
  (hydraulic routing) is usually necessary due to
  the nature of the flow, which is rapidly
  changing.

76
Factors Affecting the Hydrologic Response

• Current Conditions
• Precipitation Patterns
• Land Use
• Channel Changes
• Others..

77
Current Conditions

• Wet
• Dry
• Update model states
• subjective

78
Precipitation Patterns

• The pattern is both temporal and spatial.
• A storm moving away from an outlet will have a
  very different result than the identical storm
  pattern (spatially) moving towards the outlet.
• Lumped hydrologic models have a very difficult
  time in simulating spatially and temporally
  varied storm events.
• The very nature of MAP values - indicates one
  of the problems.
• A forecaster must understand the potential of
  precipitation patterns to affect the forecast

79
Land Use

• Urban
• Agricultural
• Anything that changes the infiltration, runoff,
  etc...

80
Channel Changes

• Slopes
• Storage
• Rating Curve
• Ice!!!

81
Rating Curves

• Rating curves establish a relationship between
  depth and the amount of flow in a channel.