web viewthe instructor explains locale specific logistics, in particular location of restrooms....

Texas Department of Transportation

DES 601 – Basic Hydrology and Hydraulics

Human Resources DivisionWorkforce Development Section

Instructor Guide

Revised August 31, 2013

DRAFT

Table of Contents

Hour 1 — Introduction...................................................................................................3Section 1 — Course Objectives.......................................................................................3Section 2 — Course Overview.........................................................................................3Section 3 — Course Assessment....................................................................................4Section 4 — Course Relationship to TxDOT Training Program.......................................4Section 5 — Lesson Summary.........................................................................................5Section 6 — Icon List.......................................................................................................5Hour 2 — Hydrology and Hydraulics............................................................................8Section 1 — Definitions....................................................................................................8Section 2 — Hydrologic Cycle..........................................................................................8Section 3 — Hydrology and Infrastructure.....................................................................10Section 4 — Data Requirements....................................................................................12Section 5 — Lesson Summary.......................................................................................12Hour 3 — Flood Frequency.........................................................................................13Section 1 — General Concepts......................................................................................13Section 2 — Models.......................................................................................................14Section 3 — Flood Frequency Curve.............................................................................15Section 4 — Plotting Position.........................................................................................16Hour 4 — Streamgage Analysis I................................................................................18Hour 5 — Streamgage Analysis II...............................................................................23Hour 6 — Regional Frequency Analysis....................................................................24Hour 7 — Flood Runoff Modeling I.............................................................................25Section 1 — Flood Processes........................................................................................25Section 2 — Computational Methods.............................................................................26Section 3 — Rational Method........................................................................................28Section 4 — Unit Hydrograph Based Methods...............................................................28Hour 8 — Flood Runoff Modeling II............................................................................29Section 1 — Design Rainfall Hyetographs.....................................................................29Section 2 — Rainfall Loss Models..................................................................................31Section 3 — Basin Response Models............................................................................33Hour 9 — Flood Runoff Modeling III...........................................................................35Section 1 — Watershed Subdivision..............................................................................35Section 2 — Channel and Reservoir Routing.................................................................36Hour 10 — Hydraulic Continuity Concepts................................................................40Section 1 — Conservation of Mass................................................................................40Section 2 — Area and Velocity......................................................................................40Section 3 — Geometric Properties.................................................................................40Hour 11 — Energy and Momentum Concepts...........................................................42Section 1 — Potential, Kinetic, and Pressure Energy....................................................42Section 2 — Bernoulli Equation.....................................................................................42Section 3 — Momentum Concepts.................................................................................42Hour 12 — Closed Conduit Flow.................................................................................43Hour 13 — Open Channel Flow I.................................................................................44

DES 601 Participant Guide August 31, 20122

Section 1 — Energy and Momentum.............................................................................44Section 3 — Normal Depth............................................................................................47Hour 14 — Open Channel Flow II................................................................................49Section 1 — Gradually- and Rapidly-Varied Flow..........................................................49Section 3 — Energy and Momentum.............................................................................51Hour 15 — Channel Analysis and Design I................................................................52Hour 16 — Channel Analysis and Design II...............................................................52Hour 17 — Culverts I....................................................................................................52Hour 18 — Culverts II...................................................................................................52Hour 19 — Bridges.......................................................................................................52Hour 20 — Rating Curves............................................................................................52Hour 21 — Solids Transport........................................................................................52Hour 22 — Stormwater Management Concepts........................................................52


Module 1 — Introduction The introduction to hydrology and hydraulics course is to provide Texas Department of Transportation Engineering Technicians, Graduate Engineers (EITs), and Engineers (PE) a first-course (ET) or a refresher course (EIT, PE) in hydrology and hydraulics.

Course content is adapted from the Texas Department of Transportation Hydraulic Design Manual, FHWA HDS-02-00X, and various other public domain and copyrighted sources.

Section 1 — Instructor and Participant Background

The instructor displays Figure 1.

The instructor introduces him/herself. Explain the sign-in sheet process. Remind participants about the VTC letter (for participants who travel to attend the course). Have participants introduce themselves.

Section 2 — Training Logistics

The instructor displays Figure 2.

The instructor explains locale specific logistics, in particular location of restrooms. Short breaks every hour are anticipated. Lunch breaks adjusted for local situations (to allow time for lunch and return for afternoon modules). Instructor explains that there is an assessment at the end of the course. Instructor explains that participants must attend entire course and pass assessment for passing outcome.


Figure 1. Instructor and Participant Background

Figure 2. Training Logistics


Section 3 — Course Objectives

Instructor displays Figure 3. While displayed, instructor states

“Upon completion participants will be able to

1. Derive watershed parameters, including delineation of drainage area, watershed lengths, slopes, runoff coefficients, and NRCS Curve numbers, time of concentration, etc.

2. Derive flow rates using hydrologic methods including Rational method, Regression Equations, TR55 for simple unit hydrograph application.

3. Discuss the concepts of the continuity equation, energy concepts, weir and orifice concepts, etc. with one-dimensional, steady state applications to open channel flow; and

4. Perform a simple hydraulic analysis.”

Section 2 — Course Overview

The course covers in varying detail elements of the following topics.

Course Code: DES601

Course Duration: 24 hours (3 days)

Course CEU/PDH: 2.4 hours/24 hours

Section 3 — Course Assessment

Upon completion participants should be able to

Participant’s learning is assessed with a final examination conducted in the last hour of the course. Participants earn one of three course outcomes

Outcome Requirements RemarksPassed Attend all lessons.

Final exam score = 80% or higherEarn PDH/CEU

Attended Attend all lessons.Final exam score = 79% or lower

Do not earn PDH/CEU


Did not attend Miss any or all lessons.Final exam score = none.

Do not earn PDH/CEU

Section 4 — Course Relationship to TxDOT Training Program

This course is one of several courses available to the engineer related to the interaction of naturally occurring waters and transportation infrastructure. Figure 1 is an organization chart of the current courses in the TxDOT Training inventory and includes suggested course ordering and career level of course participants. This course (Basic Hydrology and Hydraulics) is the first course in the series. The primary objective is to establish sufficient technical background such that, when combined with on-the-job training, will enable the participant to progress through the remaining courses as appropriate.

Figure 3. Training Course Organization Chart

Section 5 — Lesson Summary

Summarize lesson content in a list. Quiz/exercise as appropriate.

Section 6 — Icon List


As used in this manual, icons are visual aids that provide information to the instructor and participant on course activities during a specific section of the curriculum. The following list of icons is used throughout the curriculum. Icon use must be consistent so that instructors become familiar with the same set of icons.



Icon Description

Group Activity/Exercise

Individual Activity/Exercise

Review

Write this information down (participant only)

Performance

Play Video Clip/DVD

Action Plan

Computer Activity/Exercise

PowerPoint Slide (instructor only)

Icon Description

Question

Critical information oradvanced troubleshooting tips (participant

only)

Information

This icon indicates an important note to point out

Draw or write this information on the board/flip chart (instructor only)

Hour 2 — Hydrology and Hydraulics The objective of this lesson is to define the broad terms “hydrology” and “hydraulics” in the context of drainage design.

Section 1 — Definitions

Hydrology is the science that addresses the physical properties, distribution, and movement of water in the atmosphere, on the surface of the Earth, and in the outer crust of the Earth. For the highway engineer, the primary focus of hydrology is the water that moves on the Earth's surface and particularly that part that ultimately interacts with transportation infrastructure (that is, highway stream crossings). A related interest is to provide interior drainage for roadways, median areas, and interchanges.

Hydraulics refers to the mechanics of liquid water in physical systems and related processes. Hydraulics includes hydrostatics (water at rest) and hydrodynamics (water in motion). For the highway engineer, the primary focus of hydraulics is the behavior of water in open channels (drainage ditches, storm sewers, and often culverts) and closed conduits (force mains from lift stations, sometimes culverts). Culvert hydraulics is a separate and substantial topic and is not a examined completely in this course. Participants are encouraged to attend the culvert hydraulics course, DES604 Culvert Design and Bridges, to learn about the details of culvert hydraulics and design.

The relation between the two disciplines is that hydrologic principles are used to estimate the volume and rate water will enter a drainage system, whereas hydraulic principles are used to analyze how that water (flow) will behave in the system of interest. Application of hydrology is used to provide answers to questions such as “how many cubic feet per second will drain into a stream from adjacent property?” application of hydraulic principles are used to answer questions such as “how deep will the water be in the stream for so many cubic feet per second of discharge?

Section 2 — Hydrologic Cycle

Water exists naturally (at terrestrial temperature and pressure) in the three phases of matter — solid, liquid, and gas. The quantity of water present varies from location to location and from time to time. Although the vast majority of the earth's water is found in the world's oceans at any particular moment, there is a constant interchange of water between the oceans, the atmosphere, and the land surface. This interchange is called the hydrologic cycle.

The practical value of the hydrologic cycle is to establish a context for hydrologic systems behavior and an understanding of the interrelated components of meteorology, topography, land use, and the engineered environment. Figure 2 is a diagram of the


hydrologic cycle that depicts the transformation of water from one phase to another and the motion of water from one location to another, representing the complete hydrologic cycle on and near the Earth’s surface.

Figure 4. Hydrologic Cycle Diagram (from FHWA HDS 2)

Beginning with atmospheric moisture, the hydrologic cycle can be described as follows: When warm, moist air is lifted to the level at which condensation occurs, precipitation in the form of rain, hail, sleet, or snow can form and might fall on a watershed. Some of the precipitated water evaporates (or sublimates) as it falls and the rest either reaches the ground or is intercepted by buildings, trees, and other vegetation. The intercepted water evaporates directly back to the atmosphere, thus completing a part of the cycle. The remaining precipitation reaches the ground's surface or onto the water surfaces of rivers, lakes, ponds, and oceans.

If the precipitation falls as snow or ice and the surface or air temperature is sufficiently cold, this frozen water will be stored temporarily as snowpack to be released later when the temperature increases and melting occurs. While contained in a snowpack, some of the water escapes through sublimation and returns to the atmosphere. When the temperature exceeds the melting point, the water from snowmelt becomes available to continue in the hydrologic cycle1.

If the precipitation falls as liquid water, then the water that reaches the Earth's surface evaporates, infiltrates into the soil, flows overland into puddles and depressions in the ground, or flows into swales and streams. The effect of infiltration is to increase soil moisture. Field capacity is the moisture held by the soil after all gravitational drainage. If the moisture content is less than the field capacity of the soil, water returns to the 1 Although snow and ice occur in Texas, the most important precipitation component for transportation hydrology is rainfall.


atmosphere through soil evaporation and by transpiration from plants and trees. If the soil moisture content exceeds field capacity, the water percolates downward to become ground water.

Surface storage resulting from small puddles and depressions is called depression storage. The part of precipitation collected in depression storage can evaporate back into the atmosphere, infiltrate into the soil, or if it fills the depressions, that portion that exceeds of depression storage flows overland until it reaches natural swales and streams. The water held in depression storage is not available for surface runoff.

Before flow can occur overland and in the natural and/or manmade drainage systems, the flow path must reach its storage capacity. This form of storage, called detention storage, is temporary since most of this water continues to drain after rainfall ceases. The precipitation that percolates down to ground water is maintained in the hydrologic cycle as seepage into streams and lakes, as capillary movement back into the root zone, or it is pumped from wells and discharged into irrigation systems, sewers, or other drainage ways. Water that reaches streams and rivers may be detained in storage reservoirs and lakes or it eventually reaches the oceans. Throughout this path, water is continually evaporated back to the atmosphere, and the hydrologic cycle is repeated.

Section 3 — Hydrology and Infrastructure

The diversity of drainage problems at the natural and engineered interface (e.g. stream crossings) is broad and includes the design of pavements, bridges, culverts, siphons, and other cross drainage structures for channels varying from small streams to large rivers. Stable open channels and storm water collection, conveyance, and detention systems must be designed for both urban and rural areas. Evaluating the impacts that future land use, proposed flood control and water supply projects, and other planned and projected changes will have on the design of the highway crossing is necessary. Additionally the engineer also has a responsibility to adequately assess flood potentials and environmental impacts that planned highway and stream crossings may have on the watershed.

In drainage engineering, the primary concern is with the surface runoff portion of the hydrologic cycle. Depending on local conditions, other elements may be important; however, evaporation and transpiration can generally be discounted. The four most important parts of the hydrologic cycle to the engineer are: (1) precipitation, (2) infiltration, (3) storage, and (4) surface runoff.

Figure 3 is a hydrograph, which is a plot (graph) of discharge at a point on a stream versus time. The peak discharge is often the important quantity in hydrologic engineering as it establishes the size of hydraulic structures (i.e. culverts) that are to convey the discharge around or through the transportation infrastructure.


Precipitation is vital to the development of hydrographs and especially in synthetic unit hydrograph methods and some peak discharge formulas where the flood flow is determined in part from excess rainfall or total precipitation minus the sum of the infiltration and storage. As described above, infiltration is that portion of the rainfall that enters the ground surface to become ground water or to be used by plants and trees and transpired back to the atmosphere.

Figure 5. Hydrograph Diagram

Some infiltration may find its way back to the tributary system as interflow moving slowly near the ground surface or as ground-water seepage, but the amount is generally small. Storage is the water held on the surface of the ground in puddles and other irregularities (depression storage) and water stored in more significant quantities often in human-made structures (detention storage). Surface runoff is the water that flows across the surface of the ground into the watershed's tributary system and eventually into the primary watercourse.

The task of the designer is to determine the quantity and associated time distribution of runoff at a given highway stream crossing, taking into account each of the pertinent aspects of the hydrologic cycle. Approximations of these factors are always necessary. In some situations, values can be assigned to storage and infiltration with confidence,


while in others, there may be considerable uncertainty, or the importance of one or both of these losses may be discounted in the final analysis.

In many hydrologic analyses, the three basic elements are: (1) measurement, recording, compilation, and publication of data; (2) interpretation and analysis of data; and (3) application to design or other practical problems. The development of hydrology at the infrastructure interface is no different. Each of these tasks must be performed, at least in part, before a hydraulic structure can be designed. How extensively involved the designer becomes with each depends on: (1) importance and cost of the structure or the acceptable risk of failure; (2) amount of data available for the analysis; (3) additional information and data needed; (4) required accuracy; and (5) time and other resource constraints.

Section 4 — Data Requirements

Discourse on data required for hydrologic engineering. In general, data requirements for a hydrologic analysis vary depending on the technology chosen for the analysis. In part, this depends on whether the watershed (drainage basin) has a streamgage, the size of the drainage area, and the type of structure or analysis required. These ideas will be developed as each analysis method is presented below.

Section 5 — Lesson Summary

Summarize the lesson in a list. Quiz/exercise as appropriate.


Hour 3 — Flood Frequency The objective of this lesson is to develop the concepts of hydrology as applied to drainage design and to present technology appropriate for making estimates of design discharges.

Section 1 — General Concepts

The hydrologic cycle was presented in Lesson 2, as was the definition of the science of hydrology. The water or hydrologic cycle represents the storage of water in the atmosphere, the land surface, and the oceans in addition to the movement of water between those three stores. The science of hydrology deals with the occurrence, distribution, and motion of water in the atmosphere, the oceans, and on the land surface.

In general, transportation engineers are interested in the flood component of hydrology. That is, for drainage design, flood events are those of greatest interest. Because the occurrence and distribution of flood events is complex and difficult (if not impossible) to analyze in a mechanistic sense, flood events are usually treated as random variables. That is, they are assumed to behave more or less randomly and the principles of statistical analysis are used to establish design events.

A random variable is a quantity that can take on a range of values, each with a particular probability. Because drainage engineers are generally interested in floods that are a particular value or more, then the probability associated with each value is the probability that value might be equaled or exceeded. This is called the exceedance probability of a particular event.

Failure can be defined as the occurrence of an event of equal or greater exceedance probability. The risk of failure, then, is the probability that an event will occur of equal or greater magnitude than the design probability. Depending on the consequences of failure (loss of service, damage of infrastructure, damage to surrounding property), an acceptable level of risk is chosen for a particular design problem. This acceptable level of risk is then used as the design exceedance probability for construction of an estimate of design discharge.

The concepts of risk and design probability are related, but different. As designers, we choose the acceptable level of risk based on the consequence of failure. We then use that estimate of risk to choose the design exceedance probability.

To satisfy the statistical requirement for independence of events, we often use what is called the annual series of events. This means we use the event of greatest magnitude occurring over a period of one year. The standard is the water year, which extends from the first of October through the end of September. It happens to coincide with the


federal fiscal year, although this is probably coincidental. Estimates of probability derived from the annual series of events are called annual exceedance probability or annual exceedance frequency. For this class, we will generally drop the “annual” and use exceedance probability or exceedance frequency with the understanding that we are discussing events from the annual series.

Another concept that is commonly used in drainage design is recurrence interval or return interval. These two terms are synonymous and refer to the expected length of time between recurrences of a particular event over the long term. It is important to note the requirement “over the long term.”

There is a mathematical relation between exceedance probability and return interval. The return interval is the reciprocal of the exceedance probability. That is, if there is a one percent exceedance probability of an event, then the return interval of that event is 1/0.01, which is 100 years. To return to the concept of return interval, this means that the expected time between events with an exceedance probability of 0.01 is 100 years over the long term. This is often misunderstood and misinterpreted to mean that if a 100-year event occurs, then it will be another 100 years before an event of equal or greater magnitude will occur again. This is incorrect!

Given that flood frequency is associated with design risk and selection of both the design standard (risk) and method for estimating the design flood, a discussion of the flood frequency curve is appropriate. The definition of flood frequency curve and development of flood frequency curves is described in subsequent sections.

Section 2 — Models

Both professionals and non-professionals often use the term model. It is the kind of term that most people have some understanding about, but that understanding is generally based on inference and not definition. There are two broad categories of models used by hydrologist and hydraulic engineers. The first is the physical model. That type of model is a scale model, or a model that is geometrically scaled to correspond to the actual thing being modeled. The physical modeling approach is most often used when detailed understanding of a hydraulic structure (such as a dam spillway or a culvert junction box) is required. Physical models can be used for a specific project or to develop synthetic design aids (such as nomographs of non-linear approximating functions) to be used to size many structures.

The more common model is the numerical model, which is based on physical or empirical mathematical relations that define cause and effect between one or more input variables, a set of model parameters, and one or more output variables. For a hydraulic design engineer, the input variables are most often a set of watershed characteristics and a level of acceptable risk. The output variable is the design discharge. The parameters depend on the particular method used to relate the input and output variables.


Numerical models are subcategorized depending on whether the input and output variables are assumed to have a distribution in probability (or not). A stochastic or probabilistic model arises when the variables have a distribution in probability space. When they do not, the resulting model (or models) is called deterministic. Finally, numerical models are also subcategorized depending on whether the relations between variables are physically based or empirical. Models with variable relations defined by fundamental physical principles (conservation of mass, energy, and momentum) are called conceptual models. Empirical models are called, well, empirical.

At this point the terminology becomes a bit messy because there are four basic types of models depending on the treatment of variables and on the relations between variables. A familiar example is the rational method, which would be classified a deterministic-empirical model. It is deterministic because the input and output variables are not treated as having a distribution in probability and empirical because the relation between variables is more empirical than based in physics. Furthermore, the classification system is useful only for discussing models and for understanding differences between models.

Section 3 — Flood Frequency Curve

A flood frequency curve (FFC) is a plot of the flood discharge as a function of exceedance frequency. An example of a FFC is shown in Figure 6. The exceedance frequency is on the abscissa (x-axis) and the magnitude is on the ordinate (y-axis). As an aside, the nonexceedance probability [equal to the complement of the exceedance probability, 1-P(x≥X)] is displayed on the upper abscissa. Statisticians often work with nonexceedance probability and many statistics texts use nonexceedance probability in all computations.

The natural question that follows from the definition of the FFC should be “How is a FFC developed?” That depends on whether data are available for analysis or not. If there are streamgaging data available, then the FFC might be developed from those measurements. If not, then methods of synthetic hydrology are required.


Figure 6. An example of a flood frequency curve.

Section 4 — Plotting Position

When streamgage data are available, it is useful to superimpose the measurements on the flood frequency curve. The problem is determination of the best estimate of the annual exceedance (or nonexceedance) probability of the event. This probability is estimated using the plotting position of the observation. The plotting position is the best estimate of the probability of the observation. The most common plotting position formula is the Weibull Formula,

P ( x≥ X )= nN+1

,

where n is the rank order of the event (sorted from largest to smallest) and N is the total number of events. The Weibull plotting position is that recommended by the authors of Bulletin 17B, but there are other plotting position formulae that can be used.

The computed plotting position and associated discharge can be plotted on the same coordinates as the fitted flood frequency curve to visually confirm that a reasonable fit


was obtained. It is especially important to examine the right tail of the distribution (and data) to assess that estimates produced by the fitted distribution reasonably represent the data used to develop the flood frequency curve.


Hour 4 — Streamgage Analysis I The term flood frequency analysis refers to the statistical analysis of an annual series of flood discharges at a site of interest. The presence of a dataset means that a streamgage (or streamgaging station) is located at (or “near enough” to) the site of interest. A streamgaging station is an installation (usually by the U.S. Geological Survey, USGS) at which records of stream stage are collected. There are two broad classes of streamgages. The first is a continuous recording station at which the stage (water-surface elevation) of the stream is recorded periodically. The period depends on the size of the watershed and is generally shorter for smaller watersheds (five minutes or less) and longer for larger watersheds (hourly is typical). The second type of streamgage is the crest stage gage, which is an installation that captures the maximum stage of the stream between visits by USGS technicians. In general, the maximum stage that occurs during each water year is recorded (no time of occurrence is available).

For both types of streamgages, measurements of stage are used with a stage-discharge curve to determine the corresponding annual maximum discharge. The stage-discharge curve is obtained in a variety of ways with the most common a series of discharge determinations made using a current meter to measure mean velocity at least 20 points on a cross section of the stream and depth soundings at each point. The product of velocity and area for each subsection is summed to provide an estimate of the discharge for the stage at the time of the measurement. There are other approaches to developing the stage-discharge curve as well. However, the current meter method remains the most common.

USGS streamgaging data can be obtained from the USGS website. The website is located at the URL http://waterdata.usgs.gov/nwis/sw (at the time of this writing2). A variety of data are available for download.

The Water Resources Council (WRC) Bulletin 17B report contains all of the details supporting development and application of the Pearson Type III distribution to the logarithms of annual maximum streamflow. That document should be part of every practicing drainage engineer’s library3. There are minimum data requirements for application of the approach and they depend on the target annual exceedance probability. The requirements are displayed on Table 1.

Table 1. Data requirements (in years) by annual exceedance probability.Target Annual Exceedance Probability Minimum Record Length (years)

0.10 (10-year) 8

2Although it is unlikely that the USGS water data website will change, it can be found through Google (or an alternative search engine) by looking for the term “NWIS” (without quotes, of course), which stands for National Water Information Service.3A PDF of the report is available at http://on.doi.gov/RQ48PE at the time of this writing.


http://waterdata.usgs.gov/nwis/sw

0.04 (25-year) 100.02 (50-year) 150.01 (100-year) 20

With such short record lengths, reliable estimation of the skew coefficient from station data is not possible. Therefore, the WRC team applied the concept of generalized skew. A generalized skew coefficient is one derived by averaging the skew coefficients from nearby streamgages with relatively long periods of record. These values were mapped on one-degree quadrangles and are included in the Bulletin 17B report. In addition, they can be derived from either the HEC-SSP or PeakFQ software. (The former is produced by the U.S. Army Corps of Engineers Hydrologic Engineering Center and the latter by USGS.)

The general approach to conducting a flood frequency analysis using streamgaging data and the Water Resources Council Bulleting 17B approach (log-Pearson Type III distribution) is:

1. Acquire and assess the annual peak discharge record.

2. Compute the base 10 logarithm of each discharge value.

3. Compute the mean, standard deviation, and (station) skew of the log flow values.

4. Compute the weighted skew coefficient from the station skew and regional skew.

5. Identify high and low outliers from the sample set.

6. Recompute the mean, standard deviation, and station skew of the log flow values with outliers removed from the sample set.

7. Compute flow values for desired AEPs.

Flood quantiles are computed using the frequency factor method,

where is the mean logarithm, is the flood peak, is the frequency factor (table

lookup or calculated4), and is the standard deviation of the logarithms. The frequency factor depends on the return interval (annual exceedance probability) and distribution skew coefficient.

4TxDOT has an Excel spreadsheet developed to compute the frequency factor. The filename of the spreadsheet is freqfac.xls and it is available for download from the TxDOT web site (http://bit.ly/S97wkA at the time of this writing).


http://bit.ly/S97wkA

The mean logarithm is computed by

where is the mean logarithm (base 10) of the annual peak discharges, is the

logarithm of the annual peak discharge, and is the number of events. The standard deviation of the annual maxima is computed using

where is the standard deviation of the logarithms of the annual maximum discharges and the other variables are as defined. Finally, the station skew coefficient is computer as

where G is the station skew coefficient and the remaining variables are as defined. The affect of skew coefficient on the flood frequency curve is displayed on Figure 7. A zero skew coefficient implies a lognormal distribution (the distribution is completely described by the mean and standard deviation of the logarithms). A positive skew will produce flood estimates greater than the log-normal distribution and a negative skew will produce flood estimates less than the log-normal distribution.


Figure 7. The affect of skew coefficient on the flood frequency curve.

Certain cases require special treatment. These situations arise if:1. The streamgage record is incomplete because flows were either too large or too

small to be measured.2. The record contains zero values (the stream was dry or the values were too

small for measurement during the entire water year).3. The record contains historic flows outside the systematic period of record.4. The record contains events that derive from different hydrologic mechanics — a

mixed population.Historic flows are those that occur either before the installation of the streamgage or after systematic monitoring of the streamgage ceases. (Many streamgages are used for a few years and then discontinued for a variety of reasons.) Bulletin 17B contains guidance on treating the flood frequency distributions for these four cases. Perhaps the most challenging problem is that of mixed populations. An example of this is when annual peaks sometimes occur in response to snowmelt and other times because of storm-event flooding.

The skew coefficient is sensitive to the period of record. Therefore, the WRC developed the idea of generalized skew for application of the log-Pearson III procedure. The term generalized skew refers to use of a skew coefficient averaged over some geographic area5. If the period of record is relatively short, say less than 25 years, then the generalized skew coefficient should be used. If the period of record is relatively long, say 75 years or more, then the station skew should be used. For values between those two end points, a weighted skew coefficient is recommended. The weighted skew coefficient is computed using

5The Bulletin 17B report contains a map of generalized skew coefficient by one-degree quadrangles for most of the United States. There are other sources for the generalized skew coefficient as well.


where is the station weighted skew, MSE is the mean squared error of the estimate of the skew coefficient, is the generalized skew coefficient, and is the station skew coefficient. For the generalized skew, the mean squared error is 0.35 in Texas (Judd and others, 1966) and 0.55 for application of Bulletin 17B estimates.

The mean squared error for the station skew is computed (approximately) using

where N is the record length and

for or

for and

for or

for

The final tests are those for high and low outliers. An outlier is a value that is “odd” in a statistical sense and can bias the estimates of the distribution parameters. The text of Bulletin 17B and the TxDOT HDM contain the procedure for testing and adjusting the dataset for high and low outliers.


Hour 5 — Streamgage Analysis II Other distributions…

Annual peak versus partial duration series…

Liars, outliers, and out-and-out liars…


Hour 6 — Regional Frequency Analysis The term regional analysis is used in a number of contexts. For TxDOT application, it refers to use of regional regression equations to estimate design floods for required recurrence intervals.

Regression equations are mathematical models calibrated using statistical methods that minimize the errors between the predicted (regression-model estimate) and observed values. For design flood estimation, the observed values are taken from flood frequency curves fit to streamgages. That is, at least one regression equation is developed for each exceedance frequency for which design estimates are desired. The predictor variables are chosen from a combination of watershed characteristics and other variables (such as streamgage location). The process involves selecting one (or many) candidate form of the regression equation and then calibrating the parameters for each candidate. The best candidates are selected based on statistical assessment of the strength of each equation.

Watershed characteristics are often used as predictor variables. The familiar measures of watershed drainage area, main stream slope, a development measure (basin development factor is one), watershed location, and watershed shape factor are commonly used. These values are generally measurable using mapping or field methods.

An example of a regional regression equation is

where Q2 is the estimate of the 2-year flood discharge (cfs), P is the mean annual precipitation (inches), S is the dimensionless main channel slope, is the OmegaEM parameter (from the HDM), and A is the watershed drainage area (mi2). Application of this equation is straightforward using estimates of the watershed characteristics and parameters from the HDM.


Hour 7 — Flood Runoff Modeling I Flood runoff modeling is the application of fundamental hydrologic principles to produce estimates of design floods so that the principles of hydraulics can be applied to design drainage structures. In the previous lesson the application of statistics to flood series was used to create estimates of flood peak discharge. That is one of several approaches to making estimates of n-year floods for hydraulic design.

In this lesson, additional technologies will be developed to provide other approaches for creating design flood estimates. Before the approaches are presented, the processes involved should be enumerated.

Section 1 — Flood Processes

Atmospheric water that condenses into small droplets forms clouds. Under proper meteorologic conditions, these droplets can coalesce into drops of sufficient mass to overcome turbulent lifting. If there are enough of these droplets, then we call the resulting outfall of atmospheric water precipitation6. Rainfall is the form of precipitation of most interest to flood prediction. As rainfall approaches the ground, it passes through a gauntlet of potential abstractions before runoff can occur.

These potential abstractions are: Evaporation, Interception, Infiltration, and Depression storage.

In general, the relative humidity during a heavy precipitation event is sufficiently close to saturation that there is little vapor deficit for evaporation to be a significant abstraction. Interception is the capture of rainfall by vegetal surfaces before it reaches the ground. Once rainfall reaches the ground, there is a potential for water to move into the interstitial pores of the soil in a process called infiltration. This process is governed by a combination of capillary action and gravity action. Finally, the ponded depth must be sufficient to overcome microtopographic depressions that tend to store water that is infiltrated or evaporated.

Of the potential abstractions, infiltration and depression storage are generally considered the most significant. The other abstractions are generally neglected for estimating floods.

Once runoff occurs on the surface of a watershed, it moves down-gradient first as surface flow (sheet flow), then is collected into relatively small rivulets (channel flow), and then collects into channels of ever-increasing size until the main stream (or main stem) of the watershed is reached.6Precipitation can also occur as sleet or snow, but these forms are generally of less interest to flood prediction.


Water that infiltrates into the soil profile is held as soil moisture unless field capacity is reached. Water in excess of field capacity generally moves lower into the soil profile. If that soil water moves below the active rooting zone of the predominant vegetation on the watershed, then there is a reasonable probability that water will move to shallow groundwater as recharge. The plants actively growing on the watershed surface use soil water to make food. Some of this water is expelled from the plant through a process called transpiration.

Bare soil is subject to evaporation of soil water. During evaporation, water changes state from liquid soil water to water vapor and leaves the soil surface. The combination of soil water evaporation and transpiration is called evapotranspiration and is an important component of the water balance (hydrologic cycle). However, evapotranspiration is of less importance in constructing estimates of flood events for drainage design.

Section 2 — Computational Methods

Flood frequency analysis using streamgaged data was presented in a previous section. The focus of this segment of course materials is on the use of rainfall data — usually based on statistical analysis of rainfall events — with a transform function to convert the design rainfall to a peak discharge (design discharge) or a flood hydrograph (for both peak discharge and runoff volume).

There are three components to the statistical analysis of rainfall. They are the depth of rainfall, the duration of the storm event, and the intensity of the rainfall event. That is, the meteorologist analyzes the distribution of these three rainfall variables. The tendency is for rainfall intensity to increase with decreasing duration. That is, shorter storm events of a given frequency tend to be more intense than longer duration events.

For application, we usually assume that the critical storm event is one that is about the same as the characteristic response time of the watershed. The most often used time response characteristic is the time of concentration. The time of concentration is defined to be the length of time from the beginning of the runoff event to the time when the watershed is fully contributing. That is, when all portions of the watershed that can contribute runoff to the outlet of the watershed do. An alternative definition is the time for a unit of runoff to travel from the most hydraulically-distant part of the watershed to the outlet.

Time of concentration is computed usingt c=t ov+t ch ,

where tc is the time of concentration, tov is the overland flow time, and tch is the channel flow time. The overland flow time is computed using the Kerby equation,


t ov=K (L×N )0.467 S−0.235 ,

where K is a units conversion coefficient, L is the overland flow length, N is a dimensionless retardance coefficient, and S is the dimensionless overland slope. The channel flow time is computed using the Kirpich equation,

t ch=K L0.770 S−0.385 ,

where K is a units conversion coefficient, L is the channel length (net — length to divide less the overland flow length), and S is the channel slope. These computations are documented in greater detail in the Hydraulic Design Manual, including the parameter values to be used.

Insert example here…

A final note on time of concentration is that many practitioners use the main stream extended to the watershed divide to provide parameters for both the channel and overland flow time components. An alternative interpretation is that the two quantities are not so deterministic. The intent is to arrive at an estimate of the time for the watershed to become fully contributing. There could be other locations along the watershed highlands that have flatter slopes and longer overland flow distances than along the main channel. These areas could be sampled and estimates of overland flow time be made for several locations. A representative value could be selected from that group, based on the analyst’s experience.

Storm events that last less than the time of concentration of the watershed produce reduced hydrographs because not all parts of the watershed contribute to the runoff hydrograph (the storm is too short). Storm events exceeding the time of concentration of the watershed produce peak discharges that are too small because the average rainfall intensity is less than a shorter event.

A critical assumption is that the rainfall exceedance frequency is equal to the flood runoff frequency. In general, this probably is not true because it is often observed that a heavy rainfall event produces little runoff and a relatively frequent rainfall event can produce a heavy runoff event. There is a second factor involved and that is the wetness condition of watershed soils. However, the assumption is that watershed soils are at some “average” wetness condition for the design because analyzing all of combination is not generally feasible or warranted.

Those are the assumptions. There might be exceptions, but it is standard of practice to make that assumption. Of course, it isn’t possible to choose a storm duration exactly equal to the time of concentration, so engineering judgment is required.


Section 3 — Rational Method

The rational method is used for relatively small watersheds for small drainage projects. The assumption is that average rainfall intensity for the watershed time of concentration applied to the drainage area produces a potential rate of runoff. That rate of runoff is reduced for watershed conditions using a runoff coefficient that is between 0 and 1. A runoff coefficient of unity implies that all rainfall becomes runoff. A runoff coefficient of zero implies that all rainfall is lost to hydrologic abstractions.

The rational method is defined by

Q=CIAZ,

where Q is the discharge, C is the runoff coefficient, I is the average rainfall intensity for the exceedance frequency and time of concentration, A is the drainage area of the watershed, and Z is a units conversion factor. Tables of runoff coefficients are provided in the Hydraulic Design Manual.

Although it is rarely justified, an average (representative) runoff coefficient for a watershed can be computed if a variety of land-use/land-cover types are present. The straightforward means to do that is to construct an areally-weighted average,

Cw=∑j=1

n

C j A j

∑j=1

n

A j

,

where Cw is the weighted runoff coefficient, Cj is the runoff coefficient for area j, and Aj is the area of area j.


Section 4 — Unit Hydrograph Based Methods

A unit hydrograph (sometimes called a unitgraph) is a hydrograph of runoff from a watershed with a runoff volume of one watershed unit7. If contemporaneous streamgage and rainfall data are available from a few heavy events, then a watershed-specific unit hydrograph can be extracted from the measurements. However, for most analyses site-specific data are unavailable and synthetic methods are required.

7Some texts use one watershed inch, but that sometimes leads to confusion when the unit hydrograph is used to compute a runoff hydrograph.


Hour 8 — Flood Runoff Modeling II

In addition to the duration of the design storm and the design depth, application of the unit hydrograph method requires a time distribution of rainfall. Natural storms have a near infinite set of temporal distributions. It is not possible to test all potential distributions. Therefore, a few temporal distributions were developed that should present the critical combination of events.

Section 1 — Design Rainfall Hyetographs One approach was developed by NRCS scientists. The NRCS Type II and Type III distributions are used in the region encompassed by Texas. These distributions were developed using a balanced hyetograph approach, in which the central portion of the “storm” includes the intensity for the shortest part of the depth-duration-intensity relation for the region (the five-minute event), the next interval at the center of the includes the commensurate intensity from the next-greatest part of the depth-duration-intensity relation, and so forth. The resulting temporal distributions are displayed on Figure 8. These particular distributions are based on the 24-hour event, but 6-hour distributions are also available.

Figure 8. NRCS Type II and Type III temporal rainfall distributions.


A criticism of balanced hyetographs is that they are unreasonably steep in the center portion of the event and result in rainfall intensities unlikely to be realized by actual heavy storms of equivalent duration.

There are other options presented in the Hydraulic Design Manual. These include development of a balanced hyetograph using a feature of HEC-HMS (the generalized flood hydrograph program developed by the U.S. Army Corps of Engineers Hydrologic Engineering Center) and distributions developed through the TxDOT research program using Texas-specific data. An example of the latter is presented in Figure 9.

Both the NRCS and the balanced hyetographs are embedded in HEC-HMS. Application of Texas-specific technology requires hand or spreadsheet computations and then input of the resulting hyetograph into the HEC-HMS software.

Figure 9. Dimensionless hyetographs for Texas storms from 0- to 72-hour duration.


Section 2 — Rainfall Loss Models The next process is the loss of rainfall attributable to hydrologic abstractions. For heavy (flood-producing) storm events, it is unlikely that either evapotranspiration or interception is significant or observable. Therefore, the principal loss is attributable to a combination of depression storage and infiltration into the soil matrix. The result from subtracting these abstractions is called excess or effective rainfall. It is equivalent to storm runoff.

There are many approaches to computing excess or effective rainfall from incoming rainfall. Four main approaches (documented in the Hydraulic Design Manual) are the initial and constant-rate loss model, the Texas initial and constant-rate loss model, the NRCS curve number rainfall-runoff model, and the Green-Ampt infiltration function.

Perhaps the simplest approach from an applied perspective is the NRCS curve number method. The method uses a single parameter, the curve number, to represent the runoff-producing potential of watershed soils. Usually an areally-weighted representative curve number is developed using soil maps and land-use/land-cover maps to access tables of curve numbers. The resulting curve number is then used for rainfall-runoff computations. Components of this approach are hydrologic soil group, a classification of soils according to runoff-producing potential (with HSG A being the least likely to produce significant runoff and HSG D being the most likely). HSG is determined from NRCS publications (paper and digital). Land-use/land-cover is used to access the table values published in the Hydraulic Design Manual.

A climatic adjustment is possible if the analyst thinks it important. TxDOT research projects determine that there is a trend to reduced curve numbers from the east to the west. One approach is displayed on Figure 10. An alternative is included in the Hydraulic Design Manual.


Figure 10. Climatic adjustment factors for NRCS curve numbers.

The initial and constant-rate loss model was calibrated for Texas watersheds. The model is represented by

f (t )=I (t ) ,

where f(t) is the infiltration rate, I(t) is the rainfall rate, and P(t)<Ia, where P(t) is the precipitation and Ia is the initial loss. That is, all rainfall is lost to the initial loss until the initial abstraction is satisfied. Once the initial loss is satisfied, then the loss rate is

f ( t )=I ( t )−L,or

f ( t )=L ,


where L is the constant-loss rate. The upper form is used if the rainfall rate exceeds the constant loss rate and the lower form when the rainfall rate is insufficient to satisfy the constant loss rate.

The parameters (initial loss and constant-loss rate) are estimated using

I a=2.045−0.5497 L−0.9041−0.1943D+0.2424 R−0.01354CN ,

where L is the main channel length, D is a development parameter (0 for undeveloped watersheds, 1 for developed watersheds), R is a “rocky” parameter (0 for non-rocky watersheds, 1 for rocky watersheds), and CN is the NRCS curve number. The constant loss parameter is computed using

CL=2.535−0.4820L0.2312+0.2271R−0.01676CN ,

where the regression parameters are as for the initial loss equation. These values can be directly used in HEC-HMS for converting incoming rainfall to runoff for the unit hydrograph procedure.

Section 3 — Basin Response Models In general, a unit hydrograph is a basin response model. The watershed’s unit hydrograph represents the time response of the watershed to a unit pulse of runoff. That is, the unit hydrograph accommodates watershed transit time and storage as runoff moves from the most distal point of the watershed to the outlet.

A common synthetic unit hydrograph is the NRCS dimensionless unit hydrograph. The curve is shown on Figure 11. There are other forms of the unit hydrograph that can be used and these are documented in the Hydraulic Design Manual.


Figure 11. NRCS dimensionless unit hydrograph.

A unit hydrograph for a specific watershed depends on a number of watershed characteristics. For application of the NRCS dimensionless unit hydrograph, there are only two watershed characteristics required — the watershed drainage area and the watershed time of concentration. Given these two characteristics, the peak discharge, Qp, and the time to peak, tp, are computed using

t p=Δt2

+t l ,

where Δt is the computational time interval and tL is the lag time for the watershed. The computational time interval is computed using

Δt=0.133t c ,

where tc is the time of concentration of the watershed. The computational time interval should be adjusted to something reasonable (even multiples of minutes, for example). The lag time is taken to be three-fifths of the time of concentration,

tL=0.6 t c .Finally, the unit hydrograph peak discharge, Qp, can be computed using

Q p=C f KAt p

,


where Cf is a conversion factor (645.33), K=0.75 (geometric constant for the shape of the curvilinear unit hydrograph), A is the drainage area, and tp is the time to peak discharge. This equation can be simplified by substitution of the constant parameters,

Qp=484 Atp

,

where the variables are as previously presented. The constant, 484, represents a unit hydrograph with 3/8 of the area under the rising limb of the hydrograph. If the watershed is particularly steep, this parameter can increase to approximately 600. If the watershed is very flat, then the peak is substantially delayed and the parameter can be as little as 300. Unless specifically warranted, the standard value of 484 should be used for Texas watersheds.


The NRCS curvilinear unit hydrograph was chosen as a common example of the development and use of a unit hydrograph for flood runoff computations. The procedure is embedded in a commonly used computer program, HEC-HMS, and is considered useful for application to Texas watersheds. There are a number of other synthetic unit hydrographs that can be used for Texas watersheds. These are documented in the Hydraulic Design Manual.

Timing – this needs something, but I’m not sure exactly what just now…

Hour 9 — Flood Runoff Modeling III Section 1 — Watershed Subdivision Based on results published in Research Report 5822–01–2, watersheds should not be subdivided with the assumption that subdividing the watershed will result in more “accurate” flood hydrographs. Unnecessary watershed subdivision results in a proliferation of parameters that must be estimated. The uncertainty in the parameter estimates overshadows the implied improvement in modeling results. Therefore, watersheds should only be subdivided when there is a compelling reason. Such reasons might be substantial differences in watershed characteristics or the presence of regulating structures on a major portion of the watershed that will influence the arrival time of a significant portion of the flood hydrograph.

If a watershed is subdivided, then mechanics to “move” the hydrographs from subwatersheds to the watershed outlet are required. Two types of hydrograph routing are used in routine drainage design. If the hydrograph is to be moved along the channel network, then a channel routing algorithm is required. If a pond, reservoir, or other


detention facility is located along the drainage path, then reservoir routing might be needed.

Section 2 — Channel and Reservoir Routing A sketch of the movement of watershed through a watershed is shown on Figure 12. Runoff as expressed by the hydrograph at an upstream point (Point A) as it moves to a downstream point in the watershed (Point B) is depicted in the sketch. The effect of hydrograph routing is to delay the peak discharge (the travel time), attenuate (reduce) the peak discharge, and disperse the runoff hydrograph about the peak discharge.

Figure 12. The impact of hydrograph motion through a watershed.

The simplest approach to both routing problems is the use of a hydrologic routing method. The more complex approach is a hydraulic routing method. The difference between hydrologic and hydraulic routing methods is that hydrologic methods are based on conservation of mass only. Hydraulics methods require application of both conservation of momentum (or energy) and conservation of mass. Therefore, hydraulic methods are inherently more complicated, more difficult to apply, and require more resources. Use of a hydraulic method should be fully justified before selection of that approach.

The unsteady state conservation of mass equation is expressed as


I−O=∆ S∆ t,

where I is the inflow rate (cfs or cms), O is the outflow rate (cfs or cms), and ∆ S /∆ t is the rate of change of storage with respect to time (cfs or cms).

The Muskingum channel routing method is the most straightforward of the channel routing approaches. The Muskingum algorithm is based only on conservation of mass and the assumption that channel storage can be estimated as two components — “wedge” storage and “prism” storage. Prism storage is that part of water in the channel between two computational sections for a constant depth of flow. Wedge storage is that part of the water in the channel above the prism storage. That is, total storage in the channel computational segment is

S=K [ XI+(1−X )O ] ,

where K (seconds) is the wave travel time, and X (dimensionless) is the weighting factor used to describe storage in the channel reach. K is often approximated as the average travel time through the reach. K can be approximated using the channel velocity through the reach. The value of X can range from 0 (pure translation of the wave without attenuation) to 0.5 (maximum impact of channel storage on the routing algorithm). The value of X for natural channels is between 0.0 and 0.3, with 0.2 being a commonly used value8.

The storage equation and mass equation are combined and simplified to produce the Muskingum routing equation,

Ot+1=C1 I t+1+C2 I t+C 3Ot ,

where C1, C2, and C3 are the routing parameters, It+1 is the inflow rate at the new time step, It is the inflow at the current time step, and Ot is the outflow at the current time step. The routing parameters are computed using

C1=∆t−2KX

2K (1−X )+∆ t,

C2=∆ t+2KX

2K (1−X )+∆ t,and

C3=2K (1−X )−∆t2K (1−X )+∆ t

.

8Traditionally, the routing parameters were determined by calibration using measured hydrographs.


Finally, the time step, ∆ t , should be between 0.3K and K for proper numerical behavior and the number of routing reaches should be about

n= K∆ t.

Use of the number or routing reaches from the equation above and the time step in relation to the travel time should produce a numerically-stable solution.

Reservoir routing is usually computed using a hydrologic technique. The modified Puls method is based on an assumption that the reservoir pool is always level. That assumption forces the stage-storage curve to be single valued (no wedge storage), which simplifies the derivation and application of the technique. The conservation of mass equation can be discretized,

I t+I t+¿

2−Ot+Ot+1

2=S t−S t+1∆ t

, ¿

where the two terms on the left-hand side are the inflow and outflow terms at the middle of the computational time step and the right-hand side is an approximation of the time derivative of storage. The inflow rate is specified for all time steps. The outflow rate and the storage are both known at the current time step (by previous computation). Therefore, if all the known terms are placed on the right-hand side and the unknown terms on the left hand side, then

2St+1∆ t

+Ot+ 1=I t+ I t+1+2S t∆ t

−Ot ,

where the left-hand side is called the “storage indication” of the reservoir. This term is pre-computed using the level-pool assumption as a function of reservoir stage. The outflow-stage relation is also pre-computed and is used to determine the outflow at the end of the computational time period. In some presentations, the outflow rate and storage indication are combined on the same graph (Figure 13 for example) and the outflow is determined directly from the storage indication. Then, given the values on the right-hand side of the equation, the left-hand side is computed, the elevation for the storage indication determined using a table look-up or linear interpolation, and the outflow from the reservoir determine. The computation is advanced using the new values computed to compute the right-hand side of the routing equation and continuing in a step-wise fashion.


Figure 13. An example of a storage indication versus outflow rate curve for level-pool reservoir routing.


Hour 10 — Hydraulic Continuity Concepts

Section 1 — Conservation of Mass

The single most important principle of fluid mechanics is conservation of mass. For incompressible flows, the volumetric flow rate can be used in place of the mass flow rate,

I−O=dSdt,

where the variables are as previously defined. If the flowrates are unchanging with respect to time — the flow is steady state — then the time derivative is zero and the inflow and outflow rates are equal. Because Q=VA , that is, the discharge is the product of mean velocity of flow and the flow area, then the principle of continuity for steady flows is that the product of velocity and area are constant. This fact can be used to solve a variety of problems.

Section 2 — Area and Velocity

In the previous section, the concepts of mean velocity and flow area were used to describe steady-state conservation of mass, commonly called the continuity principle. The flow area of a fluid is the cross sectional area for a section constructed perpendicular (normal) to the direction of flow. That is, all velocity components of the fluid field are perpendicular to the section representing the flow area.

Although the mean velocity of flow is used to represent the fluid velocity, in actuality there is a distribution of velocity over the flow area. Near the flow boundary, the fluid velocity is near zero (the law of the wall). Depending on the shape of the channel or conduit, the flow velocity is near maximum at the center of the conveyance. The mean velocity is the ratio of the volumetric discharge and the flow area. There are mechanics for construction of the mean velocity using integration, but those approaches are interesting from a theoretical standpoint yet not very useful for most practical applications. The exception is streamgaging, which is a form of numerical integration over the cross section of an open channel to arrive at an estimate of the volumetric discharge.

Section 3 — Geometric Properties

Other geometric properties are useful and required for many computations. The flow area was presented in the previous section. Three other properties are frequently used in hydraulics. They are the hydraulic radius, the hydraulic depth, and the wetted perimeter of flow. The wetter perimeter is the perimeter of the wetted section of the flow area. For example, for a circular pipe flowing full, the wetted perimeter is

P=πd ,


where P is the wetted perimeter, π is the usual constant, and d is the diameter of the pipe. That is, the wetted perimeter of a full circular pipe is the circumference of the pipe.

The top width of a section, T, is the width of the free water surface (obviously this refers to an open channel) along the cross section where the flow area is measured or computed. For a rectangular flow area, the top width is the width of the channel. The top width is used in the definition of the hydraulic depth,

D= AT,

where D is the hydraulic depth, A is the flow area, and T is the top width of the channel.

The final useful geometric property is the hydraulic radius, R. The hydraulic radius is used in both open channel and closed conduit flows. It is defined as

R= AP,

where R is the hydraulic radius and the other variables are as previously defined. It is interesting to observe that the hydraulic radius of a circular pipe is

R=

π4d2

πd=d4,

where the variables are as previously defined.


Hour 11 — Energy and Momentum Concepts Section 1 — Potential, Kinetic, and Pressure Energy

Potential energy is the energy attributable to position. Because gravity is the main source of energy in fluid flows, then the elevation of a fluid’s center of mass represents the energy of position of the fluid mass, or the potential energy.

Kinetic energy is the energy of motion. The kinetic energy of a fluid mass is related to the square of the mass’ velocity.

Finally, pressure energy is the energy stored in a contained fluid by compression of the fluid (and tension in the container). Even for incompressible flow, substantial pressure energy can be stored in the fluid with a slight compression of the fluid.

These three energy components can be combined to describe conservation of energy. That is, the sum of potential energy, kinetic energy, and pressure energy must be conserved.

Section 2 — Bernoulli Equation

The conservation of energy principle from the previous section can be written as an equation,

pγ+ V

2

2g+z=K ,

where p is the pressure, γ is the specific weight (weight density) of the fluid, V is the mean velocity, g is the gravity constant, z is the height of the center of fluid mass above an arbitrary datum, and K is a constant (the total energy of the flowing mass). This equation was developed long ago and is called the Bernoulli equation. It describes conservation of energy in an ideal fluid and can be used to solve many simple hydraulic problems in which energy losses are negligible.

Section 3 — Momentum Concepts

The momentum of a flowing fluid is the product of the flowing mass and velocity. Momentum is a vector quantity, unlike energy. Therefore, direction of flow is important. Fluid momentum is described by

∑ F= ρQ∆V ,

where F represents one (or more) external forces acting on a fluid volume, ρ is the fluid density, Q is the volumetric flowrate of the fluid, and ∆V is the vector change in velocity of the fluid volume. Stated in words, fluid momentum is conserved the vector sum of the


external forces acting on a fluid volume is equal to the product of the fluid density, the flowrate, and the vector difference in fluid volume velocity.

The momentum equation is useful for the solution of certain hydraulic problems, particular in estimating the forces in pipe bends and for designing thrust blocks. It is also used in computing the flow through a hydraulic jump. The latter is to be discussed in the section on open channel flow.

Hour 12 — Closed Conduit Flow In a real fluid, there are losses (actually conversions) of energy in moving fluid mass from one location to another. Therefore, a more general representation of the Bernoulli equation is

p1γ

+V 12

2g+z1+hp=

p2γ

+V 22

2g+z2+hf ,

where the numerical subscripts denote the locations where the energy equation is written, hp is any energy (head) introduced by a pump in the system and hf is the energy lost (converted to heat) by friction. The Manning equation is often used to relate the energy loss to geometric properties of the hydraulic system,

Q=0.4644n

d8 /3S1/2 ,

where n is the Manning friction loss coefficient, S is the slope of the energy grade line (the energy slope), and the constant 0.4644 is a combination of units conversion and change from hydraulic radius to pipe diameter9. The energy slope is the ratio of the head lost to friction and the length over which that energy is dissipated, S=hf /L. The Manning equation can be solved for the energy loss and that solution used in the energy equation,

p1γ

+V 12

2g+z1+hp=

p2γ

+V 22

2g+z2+L( nQ

0.4644d8 /3 )2

,

where the variables are as previously defined. The notion of total head is the use of all terms of the energy equation, as included in the previous equation. Total head is useful whenever the friction loss and velocity head are significant parts of the total available energy in the system.

9Use of four significant figures is inappropriate. Manning’s n is generally known to only two significant figures. The remaining components of the Manning equation are probably known to three significant figures at most.


Two additional concepts are useful. The first is the hydraulic grade line (HGL). The HGL represents the line along the conduit to which water would rise in a piezometer. The HGL is defined as the sum of potential and pressure energy,

H=z+ pγ.

The second is the energy grade line. The energy grade line represents the total energy in the flow in the conduit,

E=z+ pγ+ V

2

2 g.

The difference between the HGL and the HGL is the kinetic energy (velocity head) of the flow. For flow in a conduit of constant geometry (constant diameter, for example), the HGL and the EGL are parallel.

Because the momentum principle is more challenging to work with, most fluids problems are solved using conservation of energy (the Bernoulli equation). However, there are applications for which forces are important. Those are instances for application of the momentum equation. The conservation of momentum equation for closed conduits is

∑ F= ρQ∆V ,

where F is each of the vector-valued external forces, ρ is the fluid density, Q is the volumetric discharge rate, and V is the vector velocity. In words, this principle is the vector sum of external forces acting of a fluid mass is the product of the fluid density, the volumetric flow rate, and the vector change in velocity.

Hour 13 — Open Channel Flow I

Section 1 — Energy and Momentum

An open channel occurs when the water has a free surface. That is, the conveyance does not completely enclose the flow. As a direct result, the hydraulic grade line of flow in an open channel is at the free surface. It is the location of the sum of the potential energy and the pressure energy.

For open channels, the reference point for pressure energy is usually the bottom (thalweg) of the channel. Therefore, the elevation of the HGL is the sum of the channel bottom elevation and the depth of flow.


As for closed conduit flow, the mean velocity of open channel flow is the ratio of the volumetric discharge to the flow area, that is

V=QA.

As for closed conduit flow, the flow area is perpendicular (normal) to the main direction of flow. The kinetic energy of flow is expressed as the velocity head, V 2/2g.

The energy equation is also useful for open channel flows. However, unlike closed conduits, the reference point is the channel bottom or thalweg. So, the energy equation becomes the specific energy equation,

E=d+α V2

2g,

where d is the flow depth, α is the velocity head correction coefficient10, and the velocity head is as previously described. The total energy, then, is the sum of the bottom elevation and the specific energy. Another way to express the total energy is the sum of the water-surface elevation and the velocity head.

If the volumetric discharge is held constant and the depth allowed to vary11, a plot of specific energy versus depth can be constructed. Such a graph is presented in Figure 14. Some terms are presented in Figure 14 that require explanation. The specific energy curve has a minimum value. The minimum specific energy occurs at critical depth. The exact formulation for determining critical depth can be obtained by differentiating the specific energy equation with respect to depth and then solving for the critical value (where the derivative is equal to zero).

Critical depth occurs at minimum specific energy. This represents the minimum amount of specific energy that can exist for the combination of channel geometry and discharge. If the depth of flow is less than critical depth, then the velocity head is relatively large and the flow is dominated by the kinetic energy. This flow state is called supercritical flow (some texts use rapid flow). Conversely, if the depth of flow exceeds critical depth, then the pressure energy (depth energy) dominates and the flow state is called subcritical flow (and some texts use tranquil flow).

10The kinetic energy coefficient is approximately one for simple channels. It can be significantly greater than one (1.3 to 1.5) for compound channels. This is an advanced topic and details are presented in the Hydraulic Design Manual.11If depth is varied, then so is the flow area and velocity. So, there is a trade-off between depth and velocity head.


Figure 14. Specific energy diagram for a hypothetical channel.

The importance of critical flow and critical depth to analysis of open channel flow cannot be over-emphasized. The concept of hydraulic control is a location where the depth of flow is forced to reach or pass through critical depth. Hydraulic controls occur where the channel slope changes from subcritical to supercritical, at overflow weirs, and at strong contracts in the channel (such as culverts and bridges). If the hydraulic control can be identified, then critical depth can be computed at that location and be used to determine how flow in the channel and at the location will occur.

The Froude Number is a useful concept for the analysis of open channel flow. The Froude number is usually denoted as F and it is a dimensionless number that relates the ratio of momentum to gravitational forces. It is expressed as

F= V√gD

,

where D is the hydraulic depth and the other variables are as previously defined. The denominator represents the celerity (velocity) of a small gravity wave in the flow. Therefore, the Froude number relates the mean flow velocity to the wave celerity in the channel. If the Froude number is unity, then the flow is critical. If it is less than one, the flow is subcritical, and if the Froude number exceeds one, the flow is supercritical.


An alternate approach is the use of the conservation of momentum principle. As applied to open channels, the most common form of the momentum equation is called specific force, and is

F=Q2

gA+z A,

where F is the specific force, z is the depth to the centroid of the flow area, and the remaining variables are as previously defined. If specific force is conserved, then this principle can be used to compute the change in depth through a rapidly-varied section.

The specific force equation can be solved for the ratio of the sequent depths through a hydraulic jump in a rectangular channel,

d2d1

=12 (−1+√1+8F12 ) ,

where d2 is the sequent depth after the hydraulic jump, d1 is the initial depth, and F1 is the initial Froude number. It should be clear that the initial Froude number must exceed one (be supercritical) or there is no hydraulic jump.

Section 2 — Steady-state, Uniform Flow

If the volumetric flowrate does not vary with respect to time, then the flow is called steady state. Conversely, if the flowrate varies with time, then the flow is unsteady state. If the depth of flow is constant along a reach of channel, then the flow is called uniform. Uniform flow requires a prismatic channel, that is a channel with unvarying geometric properties (including slope). Although unsteady, uniform flow is possible, it is rarely observed. Therefore, when hydraulic engineers use the term uniform, steady state is implied.

Section 3 — Normal Depth

The term normal depth is associated with the depth of (steady) uniform flow. The Manning equation is appropriate for uniform flow computations in open channels. For open channels, the Manning equation is often expressed in the conveyance form,

Q=K S1 /2 ,

where Q is the volumetric discharge (as usual), K is the channel conveyance, and S is the slope of the energy grade line. The conveyance is defined as

K= znA R2/3 ,


where z is a dimensional coefficient (1.49 for English units12 and 1 for SI), n is the Manning roughness coefficient, A is the flow area, and S is the slope of the energy grade line. If the channel is prismatic (constant geometry), then it is possible (actually, it is “normal”) for the flow to seek a balance between the rate of energy addition (bottom slope) and the rate of energy dissipation (energy slope). When the energy slope and bottom slope are equal (rates in equilibrium), then the flow is called normal. The depth associated with normal flow is called normal depth and it is a useful value in the design of open channels.

For example, consider a rectangular channel. If the bottom width is b, then the flow area is give by A=db, where the depth of flow is d. The wetted perimeter is P=b+2d, so the hydraulic radius is R=db/(b+2d). The conveyance of the channel is then

K= 1.5nbd( bd

b+2d )2/3

,

where the variables are as previously presented. If normal depth is desired, then the Manning equation can be solved for the conveyance,

K= QS1 /2

,

and the resulting form of the conveyance is implicit in the depth of flow. That means that the computation of normal depth requires a numerical method (iteration, trial and error) to arrive at a reasonable estimate.

Continuing with the example, suppose the rectangular channel has a longitudinal slope of 0.001, a width of 10 ft, is concrete lined (n=0.013), and the discharge is 125 cfs. The conveyance for this configuration is

K= QS1 /2

= 1250.0011/2

=3,950 cfs.

The final step is to determine the depth of flow required to provide this conveyance. The conveyance equation is

K=3,950= 1.50.013

(10 )d ( 10d10+2d )

23=1,150d ( 10d10+2d )

23 .

12Again, given the accuracy achievable with natural (or even man-made) channels, use of 1.5 for the Manning equation and English units is sufficient.


The solution to this equation is d=2.45 ft, more or less. Therefore, the normal depth of flow for this channel and discharge is about 2.45 ft.

Hour 14 — Open Channel Flow II

Section 1 — Gradually- and Rapidly-Varied Flow

It is possible that the depth of flow varies along the channel reach. If that variation is relatively gradual, then the term gradually varied flow is used. In contrast, if the variation occurs over a relatively short distance and is substantial, then the flow is called rapidly varied. An example of a gradually-varied flow is the backwater upstream from a hydraulic control (such as a weir or culvert). Examples of rapidly-varied flow include both hydraulic jumps and hydraulic drops. A hydraulic jump occurs where a supercritical flow is forced to pass through critical depth. Because of the minimum specific energy requirement, the passage occurs over a relatively short distance as a singularity and the water surface changes abruptly. A hydraulic drop occurs at an overflow weir (for example). Although a hydraulic drop is less spectacular than a hydraulic jump, the change in depth occurs just as abruptly and there is observable curvature in the water surface.

In general, the energy principle is used for gradually-varied flow computations and the momentum principle is used for rapidly-varied flow computations. The reason is that accounting for energy losses in rapidly-varied flow is challenging and generally only approximate. However, momentum is conserved and momentum losses are generally small for rapidly-varied flow, so the computations are less subject to estimation error.

The computation of gradually-varied flow comprises a large part of open channel hydraulics. The water-surface profile is used for many design and analysis activities. Chow’s (1959) taxonomy of water-surface profiles is useful. The profiles were categorized depending on two factors. The first factor is the bed slope of the channel. The second factor is the relation of the flow depth of the profile to the normal depth and to critical depth. Definition of the water-surface profile based on the bed slope is presented in Table 2. Also include on Table 2 is the relation between normal and critical depths.

Table 2. Water-surface profile slope designations.

Channel Bed Slope Critical/Normal Depth Relation RemarksSteep — S dn < dc

Critical — C dn = dc

Mild — M dn > dc

Horizontal — H S0 = 0 dn undefindedAdverse —A S0 < 0 dn undefinded


The relation between water-surface flow profile depth, critical depth, and normal depth is presented in Table 3. Some explanation of the components presented in Table 3 is appropriate. For a Type–1 profile, the depth of flow exceeds both critical depth and normal depth. For a Type–2 profile, the depth of flow is trapped between normal and critical depths. The relation between critical and normal depths depends on the channel bed slope for a Type–2 profile. Finally, for a Type–3 profile the flow depth is less than both the normal and critical depths.

Table 3. Water-surface profile depth designations.

Profile Type Depth Relation Logic Depth RelationType–1 d > dc .AND. d > dn

Type–2 dc < d < dn .OR. dn < d < dc

Type–3 d < dc .AND. d < dn

These categories, when applied to actually channels, leads to a suite of water-surface profiles. The classic is the M1 backwater profile, which occurs when an obstruction forces an increase in depth (upstream from the obstruction). A sketch of an M1 profile is shown on Figure 15. Similarly, a sketch of an S1 profile is shown on Figure 16. Other examples are relatively straightforward to construct.

Figure 15. An example of an M1 water-surface profile. In the figure, yn and yc are used in place of dn and dc.


Figure 16. An exmaple of an S1 water-surface profile. In the figure, yn and yc are used in place of dn and dc.

Section 2 — Energy and Momentum

In general, the energy principle (specific energy) is simpler to apply. The exception to this case is computation of flow through a hydraulic jump. This is because energy losses through a hydraulic jump are difficult to estimate. Therefore, the

Hour 15 — Channel Analysis and Design I Section 1 — Terminology


Hour 16 — Channel Analysis and Design II

Hour 17 — Culverts I

Hour 18 — Culverts II

Hour 19 — Bridges

Hour 20 — Rating Curves

Hour 21 — Solids Transport

Hour 22 — Stormwater Management Concepts


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