Gradually-varied Flow Profiles in Open Channels: Analytical Solutions by Using Gaussian Hypergeometric Function

196 Pages·2014·4.63 MB·English

by  Chyan-Deng Jan (auth.)

Advances in Geophysical and Environmental
Mechanics and Mathematics
Chyan-Deng Jan
Gradually-
varied Flow
Profiles in Open
Channels
Analytical Solutions by Using Gaussian
Hypergeometric Function
Advances in Geophysical and Environmental
Mechanics and Mathematics
Series editors
Angelika Humbert, Bremerhaven, Germany
Kolumban Hutter, Zürich, Switzerland
For furthervolumes:
http://www.springer.com/series/7540
Chyan-Deng Jan
Gradually-varied Flow
Profiles in Open Channels
Analytical Solutions by Using Gaussian
Hypergeometric Function
123
Chyan-DengJan
Department of Hydraulic andOcean
Engineering
National ChengKungUniversity
Tainan
Taiwan
ISSN 1866-8348 ISSN 1866-8356 (electronic)
ISBN 978-3-642-35241-6 ISBN 978-3-642-35242-3 (eBook)
DOI 10.1007/978-3-642-35242-3
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013953191
(cid:2)Springer-VerlagBerlinHeidelberg2014
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To
Dr. Cheng-lung Chen
Preface
Many hydraulic engineering works involve the computation of surface profiles of
gradually varied flow (GVF) that is a steady nonuniform flow in an open channel
withgradualchangesinitswatersurfaceelevation.Themostwidelyusedmethods
forcomputingGVFprofilescouldbeclassifiedintothedirectintegrationmethods,
step methods (i.e., simple numerical methods), and advanced numerical methods
(e.g., the standard fourth order Runge–Kutta method). Numerical solutions of the
GVF equation are primarily used in non-prismatic channels. In some prismatic
channels, such as artificial channels, the GVF equation can be simplified so as to
let the analytical (or semi-analytical) direct integration be applied. The analytical
direct-integrationmethodisstraightforwardandcanprovidethetotallengthofthe
profile in a single computation step.
In the direct-integration method, the one-dimensional GVF equation is usually
normalized to a simpler expression in advance so as to allow the performance of
direct integration. In most cases, the GVF equation is normalized by the normal
depth, while in some cases, it is normalized by the critical depth. Many attempts
havebeenmadebypreviousinvestigatorsonthedirect-integrationmethodtosolve
the GVF equation. The table of the varied-flow function (VFF) is needed in the
Bakhmeteff-ChowproceduretoobtaintheGVFprofilesbyusingtheconventional
direct-integration method. The use of the conventional Bakhmeteff-Chow proce-
duretoobtainGVFprofileshasadrawbackcausedbytheinconvenienceoftheuse
oftheVFF-table andtheimpreciseinterpolationofthe VFF-values fromtheVFF
table. To overcome the drawback, Dr. C. L. Chen and the author had initiated a
self-investigation to look for an alternative method to get analytical solutions of
GVF profiles without recourse to the VFF since 2005. Dr. Chen was an excellent
hydrologist in the U.S. Geological Survey, and a distinguished professor in the
University of Illinois at Urbana-Champaign and in the Utah State University.
Unfortunately,Dr.ChenpassedawayinJanuary2012,justaboutthetimeatwhich
we found that the Gaussian hypergeometric functions (GHF) can be used to
analytically solve the GVF equation, and completed two drafts of manuscripts to
be submitted to a suitable journal. After some revisions by the author, these two
manuscripts were submitted to two journals and finally published, one in the
Journal of Hydrology in August 2012, and another in the Hydrology and Earth
System Sciences (HESS) in March 2013, respectively.
vii
viii Preface
ProfessorK.Hutter,theeditorofAdvancesinGeophysicalandEnvironmental
MechanicsandMathematics(AGEM2),encouragedtheauthortothoroughlywrite
outtheinnovativemethodusingtheGHFinanalyticallysolvingtheGVFprofiles
in the form of a book, when he visited our university in May 2012. The author
began to prepare this book in September 2012. This book attempts to thoroughly
introduce the innovative procedure to obtain the analytical solution of GVF pro-
filesusingtheGaussian hypergeometricfunctions(GHF)aswellastopresentthe
analysisanddiscussionsoftheGHF-basedsolutions.Thisbookisdividedintofive
chapters. Chapter 1 introduces the basic equations for the GVF in open channels.
Chapter 2 introduces the conventional direct-integration methods used to analyt-
ically solve the GVF equation. Chapter 3 presents the GVF equation normalized
by the normal depth, and the application of the GHF on analytically solving the
normal-depth-based dimensionless equation for GVF in sustaining channels. The
classification and properties of the GHF-based solutions of the GVF profiles are
also discussed in this chapter. Chapter 4 is devoted to the GVF equation nor-
malized by the critical depth, and then solves it by using GHF for GVF flows in
sustaining and non-sustaining channels. Chapter 5 presents the analysis and
classificationoftheGHF-basedsolutionsofthecritical-depth-baseddimensionless
GVFequation.Thischapteralsoshowsthatthecritical-depth-basedGVFsolution
expressed in terms of the GHF is more useful and versatile than its counterpart
normalized by using the normal depth.
TheauthorisespeciallyindebtedtoProf.C.L.Chenwhointroducedtheauthor
tothesubjectofGVFprofilesaspresentedinthisbook,aswellastothesubjectof
debrisflowwhentheauthorwasagraduatestudentintheUniversityofCalifornia,
Berkeleybetween1988and1992,andwhentheauthorwasavisitingscholaratthe
U.S. Geological Survey at Menlo Park, California, in 1999. The author also
acknowledges Prof. Y. C. Tai and Miss C. C. Shen for their help in the trans-
formation of the manuscript typed in the form of Microsoft Word to that in the
form of LaTex. Thanks are due to Prof. K. Hutter and Dr. A. Siviglia for their
reviews and valuable suggestions. The author appreciates the supports and
resources provided by the National Cheng Kung University, and the National
Science Council, Taiwan. Finally, the author would like to express his sincere
thankstoallthosewhohavedirectlyorindirectlyhelpedhiminwritingthisbook,
and to Springer Verlag and its personnel for their help in the production of the
book.
Theauthorsincerelyhopesthatthisbookwillbeareferencebookforpractical
civilorhydraulicengineerswhentheydesignhydraulicengineeringworksaswell
as for undergraduate and graduate students in the fields of civil, hydraulic, and
agricultural engineering.
Taiwan, September 1, 2013 Chyan-Deng Jan
Contents
1 Basic Equations for the Gradually-Varied Flow . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The GVF Equation for Flow in Open Channels. . . . . . . . . . . . . 1
1.3 The GVF Equation in Terms of Flow Depth. . . . . . . . . . . . . . . 4
1.3.1 Conveyance and Section Factor of Channel Section . . . . 4
1.3.2 Hydraulic Exponents Defined in Relation to Conveyance
and Section Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.3 Role of the Power-Law Flow Resistance Formula
in the GVF Equation. . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 The GVF Equation for Flow in Adverse Channels. . . . . . . . . . . 10
1.5 Classification of Gradually-varied Flow Profiles . . . . . . . . . . . . 11
1.6 Hydraulic Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.7 The Equation for GVF in Non-Prismatic Channels . . . . . . . . . . 18
1.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Conventional Integral Solutions of the GVF Equation. . . . . . . . . . 21
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 GVF Solution in Terms of Varied-Flow Function . . . . . . . . . . . 21
2.3 GVF Solution by the Bresse Method . . . . . . . . . . . . . . . . . . . . 23
2.4 GVF Solution by the Bakhmeteff-Chow Procedure . . . . . . . . . . 25
2.5 Drawbacks on the VFF Table for GVF Solution . . . . . . . . . . . . 26
2.6 Attempts Made on M and N by Previous Investigators. . . . . . . . 26
2.7 Previous Studies on Integrating the GVF Equation . . . . . . . . . . 27
2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Normal-Depth-Based Dimensionless GVF Solutions Using
the Gaussian Hypergeometric Function. . . . . . . . . . . . . . . . . . . . . 37
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Normalization of the GVF Equation . . . . . . . . . . . . . . . . . . . . 38
3.3 GVFSolutionsbyUsingGaussianHypergeometricFunctions . . . 39
3.3.1 An Alternative Form of (3.5) for juj [1. . . . . . . . . . . 40
3.3.2 Feasible Arrangement of Two Integrals. . . . . . . . . . . . . 40
ix
x Contents
3.3.3 Gaussian Hypergeometric Functions . . . . . . . . . . . . . . . 42
3.3.4 The GHF-Based Solutions of GVF Equation . . . . . . . . . 43
3.4 Alternative Method to Get the GHF-Based Solutions. . . . . . . . . 47
3.5 Classification of GHF-Based Solutions. . . . . . . . . . . . . . . . . . . 49
3.5.1 M, C and S Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5.2 An Example of GHF-Based GVF Profiles
with Specified Hydraulic Exponents . . . . . . . . . . . . . . . 50
3.5.3 N-values for Fully Rough Flows in Wide Channels. . . . . 51
3.5.4 GHF-Based Solutions Under Specified Boundary
Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.5.5 Examples of GHF-Based GVF Prolfiles
with M ¼3 and N ¼10=3, Specified k and BCs . . . . . . 56
3.6 Validation of the GHF-Based Solutions . . . . . . . . . . . . . . . . . . 57
3.6.1 Solving Equation (3.5) by Use of the ETF. . . . . . . . . . . 61
3.6.2 Comparison of the GHF-Based
and ETF-Based Solutions. . . . . . . . . . . . . . . . . . . . . . . 62
3.7 Properties of the GHF-Based Solutions . . . . . . . . . . . . . . . . . . 66
3.7.1 Slopes of Flow Profiles Varying with h =h and N. . . . . 66
c n
3.7.2 Singularities of the Rational Function Representing
the Slopes of C1 and C3 Profiles . . . . . . . . . . . . . . . . . 67
3.7.3 Points of Inflection on the M1 and M3 Profiles . . . . . . . 68
3.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.8.1 Applicability of the GHF-Based Solutions
in Perspective. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.8.2 Role of h =h in the Domain of the GHF-Based
c n
Solution Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.8.3 Reclassification of the Critical Profiles and Points
of Infinite Profile Slopes . . . . . . . . . . . . . . . . . . . . . . . 76
3.8.4 Identification of Inflection Points on GVF Profiles. . . . . 77
3.8.5 Curvature of GVF Profiles. . . . . . . . . . . . . . . . . . . . . . 78
3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4 Critical-Depth-Based Dimensionless GVF Solutions
Using the GHF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.1.1 Background and Motivation of This Study. . . . . . . . . . . 86
4.1.2 Literature Survey of GVF Profiles
in Adverse Channels . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.1.3 Objectives and Scope of This Study . . . . . . . . . . . . . . . 88
4.2 Formulation of the h -Based Dimensionless GVF Equation . . . . 89
c
4.2.1 Dimensionless GVF Equation for Flow
in Sustaining Channels . . . . . . . . . . . . . . . . . . . . . . . . 90
Contents xi
4.2.2 Dimensionless GVF Equation for Flow
in Horizontal Channels . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2.3 Dimensionless GVF Equation for Flow
in Adverse Channels . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.3 GHF-Based Solutions of the h -Based Dimensionless
c
GVF Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3.1 Alternative Forms of GVF Equations for jkvj [1 . . . . 96
4.3.2 Integration of the GVF Equation for Flow
in Sustaining Channels . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3.3 GHF-Based Solutions of GVF Profiles
in Sustaining Channels . . . . . . . . . . . . . . . . . . . . . . . . 98
4.3.4 Alternative Method to Get the GVF Profiles
in Sustaining Channels . . . . . . . . . . . . . . . . . . . . . . . . 103
4.4 GHF-Based Dimensionless GVF Profiles
in Adverse Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.4.1 Integrations of the GVF Equations for Flow
in Adverse Channels . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.4.2 GHF-Based Solutions of GVF Profiles
in Adverse Channels . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.4.3 Alternative Method to Get the GVF Profiles
in Adverse Channels . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5 Analysis of the GHF-Based Solutions of h -Based GVF Profiles. . . 115
c
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2 Analysis of the GHF-Based Solutions
of Equations (4.4) and (4.12) . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.2.1 Plotting the GHF-Based Solutions
on the ðx ;vÞ-Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
]
5.2.2 Mild (M), Steep (S), and Adverse (A) Profiles. . . . . . . . 116
5.2.3 Critical (C) Profiles. . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.2.4 Horizontal (H) Profiles in Zones 2 and 3. . . . . . . . . . . . 119
5.2.5 Horizontal Asymptotes at v¼h =h
n c
for Various h =h -values . . . . . . . . . . . . . . . . . . . . . . . 119
n c
5.3 Classification of GHF-Based Solutions Using h =h . . . . . . . . . 122
c n
5.4 Conversion from the h -Based Solutions to the h -Based
c n
Solutions, or Vice Versa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.4.1 Solutions of GVF in Sustaining Channels . . . . . . . . . . . 126
5.4.2 Solutions of GVF in Adverse Channels. . . . . . . . . . . . . 131
5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.5.1 Features of the h -Based Dimensionless GVF Profiles. . . 133
c
5.5.2 Singularities of the h -Based Slopes of the C Profiles . . . 134
c
5.5.3 Two Inflection Points of the M Profiles. . . . . . . . . . . . . 134