OPEN CHANNEL FLOW Numerical Methods and Computer Applications Roland Jeppson TECHNISCHE INFORMATlONSBiBUOTHEK UNIVERSITATSB'BUOTHEK HANNOVER Si. i. CRC Press Taylor &.Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an Informa business
List of Figures List of Tables Preface Computer Programs with Listings of Code xxiii xi xv xvii Chapter 1 Dimensions, Terminology, and Review of Basic Fluid Mechanics 1 1.1 Introduction 1 1.2 One-, Two-, and Three-Dimensional Flows 1 1.3 Steady versus Unsteady Flow 2 1.4 Uniform versus Nonuniform Flow 3 1.5 Prismatic versus Nonprismatic Channels 4 1.6 Subcritical, Critical, and Supercritical Flows 4 1.7 Turbulent versus Laminar Flow 5 1.8 Review of Basic Fluid Mechanics Principles 5 1.9 Physical Properties of Fluids and Their Effects on Open-Channel Flows 6 1.10 Conservation of Mass, or Continuity Equations 19 1.11 Energy Principle 22 1.11.1 Kinetic Energy Correction Coefficient, a 27 1.12 Momentum Principle in Fluid Flow 30 1.12.1 Momentum Flux Correction Coefficient, 3 34 Problems 38 Chapter 2 Energy and Its Dissipation in Open Channels 43 2.1 Introduction 43 2.2 Approaches to Frictional Resistance 43 2.2.1 Friction Factors in Open Channels 44 2.3 Combining the Chezy and the Chezy C Equations 52 2.4 Empirical Formula: Use of Manning's Equation 56 2.5 Channels with Varying Wall Roughness, but = Q Constant 74 2.6 Specific Energy, Subcritical and Supercritical Flows 75 2.7 Flumes 100 2.8 Delivery Diagrams 103 2.9 Graphical Aids to Solving Critical Flow Problems 131 2.10 Upstream Depth When Critical Conditions Occur at Reduced Downstream Section 145 2.11 Dimensionless Treatment of Upstream Trapezoidal Channel to Downstream Rectangular Channel 146 2.11.1 Upstream Channel Also Rectangular 150 2.12 Hydraulically Most Efficient Section 153 2.12.1 Nondimensional Variables for This Section 155 Problems 157 References 182
vi Chapter 3 The Momentum Principle Applied to Open Channel Flows 183 3.1 The Momentum Function 183 3.2 Characteristics of the Momentum Function 185 3.3 Rectangular Channels and Momentum Function per Unit Width 188 3.4 Polynomial Form for Momentum Function 194 3.5 Dimensionless Momentum Functions 196 3.6 Celerity of Small Amplitude Gravity Waves 208 3.7 Constant Height Waves 212 3.8 Open Channel to Pipe Flow 224 3.9 Multiple Roughness Coefficient for Channel Section Compound Sections 233 Problems 236 Problems to Solve Using Program CHANNEL 253 Chapter 4 Nonuniform Flows 255 4.1 Types of Nonuniform Flows 255 4.2 Ordinary Differential Equation for Gradually Varied Flow 255 4.2.1 Bulk Lateral Outflow 256 4.2.2 Lateral Inflow 257 4.2.3 Generalization of Gradually Varied Flow Equations 258 4.3 Gradually Varied Flow in Prismatic Channels without Lateral Inflow or Outflow 259 4.3.1 Classification of Gradually Varied Profiles 259 4.3.2 Sketching GVF Profiles in Prismatic Channels 264 4.3.3 Alternative Forms of the ODE That Describe GVF Profiles 268 4.4 Numerical Methods for Solving ODEs 280 4.5 Canal Systems 294 4.6 Simultaneous Solution of Algebraic and Ordinary Differential Equations 301 4.7 Flow into a Mild Channel with a Downstream Control 304 4.8 Different Modes of Gate Operation 315 4.9 Hydraulic Jump Downstream from a Gate in a Finite Length Channel 316 4.10 Nonprismatic Channels 341 4.11 Culverts 346 4.11.1 Solutions When Upstream Control Exists 347 4.11.2 Solutions When Downstream Control Exists 359 4.12 GVF Profiles in Nonprismatic Channels 368 4.13 GVF Profiles in Branched Channel Systems 374 4.14 GVF Profiles in Parallel Channels 390 4.15 Solutions to Spatially Varied Flows 399 4.15.1 Outflow from Side Weirs 400 4.15.2 Closed-Form Solution to Side Weir Outflow 404 4.16 Spatially Varied Inflows 408 4.16.1 Algebraic Solution 413 4.17 Spatially Varied Flow in Nonprismatic Channels 414 4.18 Tile Drainage 429 4.19 Downstream Controls in Nonprismatic Channels 433
vii 4.20 Gutter Flow and Outflow through Grates 449 4.20.1 Gutter Flow 449 4.20.2 Flow into Grates at Bottom of Channel 457 4.20.3 Combined Problem: Gutter Inflow and Grate Outflow 474 4.20.4 Lateral Inflow over Grate Length 484 4.20.5 No Negative Flow Rates 484 4.20.6 Last Grate at End of Gutter 492 4.20.7 Subcritical Flow through n Gutter-Grates 498 4.21 Multiple Branched Channel Systems 510 4.22 Other Dependent Variables in GVF Computations 540 4.23 Varied Flow Function 544 4.24 Moving Waves 549 4.24.1 Quasi-Unsteady Analysis 549 4.24.2 Downstream Controlled Waves 549 4.24.2.1 Explicit Method 550 4.24.2.2 Implicit Method 559 4.24.3 Upstream Controlled Waves 568 4.25 Moving Hydraulic Jump 578 Problems 593 References 656 Chapter 5 Common Techniques Used in Practice and Controls 657 5.1 Introduction 657 5.2 Resistance to Flow in Natural Streams and Rivers 657 5.3 Techniques Used for Solving Steady Flows in Irregular Channels 660 5.3.1 Defining Irregular Channel Properties 661 5.3.2 Use of Cubic Splines to Define Cross Section 667 5.3.3 Solving Manning's and Energy Equations in Natural Channels 671 5.3.4 Hand Solution to GVF-Flow by the Standard Step Method 681 5.3.5 Using an ODE Solver to Compute GVF-Profile in Irregular Channels 685 5.3.6 Multiple Parallel Channels and Flood Plain Storage 696 5.3.7 Implementation of Solution for Compound Channel 698 5.3.8 System of Natural Streams and Rivers 709 5.3.9 HEC-2 Water Surface Profiles 709 5.4 Water Measurement in Channels 710 5.4.1 Current Meters 710 5.4.2 Weirs 711 5.4.3 Parshall Flumes and Cutthroat Flumes 713 5.4.4 Other Critical Flow Flumes 718 5.5 Design of Transitions 720 5.5.1 Subcritical Transitions 721 5.5.1.1 Design of Transitions by Hand Computations 721 5.5.1.2 Transition Design by Solving an ODE Problem 724 5.5.2 Supercritical Transitions 727 5.5.2.1 Channel Contractions 728 5.5.3 Design of Supercritical Transitions 729 5.5.4 Channel Enlargements 735 5.6 Gates 741
vjjj 5.7 Submerged Flow Downstream from Vertical Gates 744 5.7.1 Dimensionless Forms of Equations 748 5.8 Series of Submerged Gates 763 5.9 Design of Side Weirs 772 5.10 Optimal Design of Trapezoidal Channels Considering Total Costs 779 5.10.1 Additional Optional Analyses 788 Problems 793 References 822 Chapter 6 Unsteady Flows 823 6.1 When Should Flow Be Handled as Unsteady? 823 6.2 Basic One-Dimensional Equations for Unsteady Channel Flows (The St. Venant Equations) 824 6.2.1 Q-Y Set of St. Venant Equations 824 6.2.2 V-Y Set of St. Venant Equations 826 6.2.3 Derivation Based Directly on Newton's Second Law of Motion 826 6.2.4 No Lateral Inflow or Outflow and Prismatic Channels 827 6.2.5 Changes in Dependent 6.3 Determination of Mathematical Type of St. Venant Equations 828 Variables 828 6.3.1 Second-Order PDEs 829 6.4 Taking Advantage of the Equation 6.5 Solution to Unsteady Flows That Deviate Only Slightly Characteristics 831 from Uniform Conditions 833 6.6 Boundary Conditions 836 6.6.1 Depth at Origin Specified, Y(0, t) Known 836 6.6.2 Velocity at Origin Specified, V(0, t) Known 837 6.6.3 Flow Rate q at the Origin Specified, q(0, t) Known 837 6.6.4 Upstream Reservoir Water Surface Specified, H(0, t) Known 837 6.6.5 Understanding Characteristics Better 851 6.7 Maximum Possible Flow Rates 863 6.7.1 Maximum Point Outflow, Aq 866 6.8 Extending the Methods to Nonrectangular Channels 867 6.8.1 Trapezoidal Channels 867 6.8.2 Dimensionless Variables 868 6.8.3 Circular Channels 883 6.9 Maximum Flow Rates in Nonrectangular Channels 890 6.9.1 Maximum Outflow AQ That Can Be Taken at an Intermediate Position 897 6.10 Positive Waves 898 6.11 Control Structures 901 6.12 Partial Instant Opening of Gates in Rectangular Channels 910 6.13 Partial Instant Closing of Gates in Trapezoidal Channels 915 6.14 Partial Instant Closure Followed by Slow Movement Thereafter 919 6.14.1 Rectangular Channels 919 6.14.2 Nonrectangular Channels 926 6.15 Dam Break Problem 937 Problems 938
i ix Chapter 7 Numerical Solution of the St. Venant Equations 963 7.1 Background 963 7.2 Method of Characteristics 965 7.3 Boundary Conditions 971 7.3.1 Downstream End 971 7.3.2 Upstream End 972 7.3.3 Solution to Boundary Unknowns 973 7.4 Using Characteristics with Specified Time Increments 987 7.4.1 Based on Second-Order Approximations 987 7.4.2 Upstream Boundary Condition Equations 989 7.4.3 Downstream Boundary Condition Equations 990 7.5 Iterative Solution Technique 1001 7.6 Explicit Evaluation of Variables at Points L and R 1003 7.7 Accuracy of Numerical Solutions 1008 7.8 Implicit Methods 1013 7.8.1 Direct Implicit Finite Differencing 1014 7.8.1.1 Boundary Conditions 1017 7.8.1.2 Solving the Difference Equations 1019 7.8.1.3 Computer Code to Implement Direct Implicit Method 1020 7.8.2 Alternative Approach to Boundary Condition Equations 1028 7.9 Gauss-Seidel or Successive-Over-Relaxation (SOR) Iterative Solution Techniques 1030 7.10 Crank-Nicolson Newton Iterative Implicit Method 1043 7.11 Weighting Current and Advanced Time Steps Differently 1059 7.12 The Preissmann Implicit Method 1064 7.12.1 Double Sweep Method of Solution 1067 7.12.1.1 Boundary Conditions 1068 7.12.1.2 Downstream Boundary Conditions 1068 7.12.1.3 Upstream Boundary Conditions 1069 7.13 Solving Preissmann Difference Equations Using the Newton Method 1075 i 7.14 Two-Dimensional Free Surface Flows 1084 7.14.1 Two-Dimensional St. Venant Equations 1084 Problems 1087 References 1100 Appendix A 1101 A.l Open Channel Geometry and Properties 1105 A. 1.1 Classification of Channels 1105 A.l.2 Geometric Properties of Common Prismatic Channels 1105 A. 1.2.1 Rectangle 1105 A.l.2.2 Trapezoid 1106 A. 1.2.3 Circle 1107 A.l.2.4 Problems (For You to Work) Obtaining Geometric Properties of Irregular Cross Sections 1114 U21
X Appendix B Numerical Methods 1123 B.l Newton Method 1123 B.l.l How the Newton Method Works 1123 B.1.2 Solving Systems of Equations 1125 B. 1.3 Implementing the Newton Method in Solving Manning's Equation 1129 B. 1.3.1 Implementation of the Newton Method in Solving Manning's Equation in a Circular Channel 1130 B.2 Lagrange's Interpolation Formula 1132 B.2.1 Implementation of Lagrange's Formula in a Computer Program 1134 B.3 Numerical Integration 1136 B.3.1 Trapezoidal Rule 1136 B. 3.1.1 Simpson's Rule 1139 B.4 Spline Functions 1146 B.4.1 Background 1146 B.4.2 Quadratic Splines 1147 B.4.3 Cubic Splines 1148 B. 5 Linear Algebra 1159 B. 5.1 Use of Subroutine LAGU That Implements Laguerre's Method 1162 Appendix C ODESOL: Subroutine to Solve ODEs 1165 C. 1 Background to Algorithm 1165 C.2 Using FORTRAN 1165 C.3 How Do You Use ODESOL 1166 C.4 Subroutine That You Must Supply 1167 C.5 odesolc-c-function to Solve ODE's 1174 C.6 How Do You Use odesolc 1175 C.7 Procedure That You Must Supply 1176 C.8 DVERK: ODE Solver from the International Statistical Mathematical Libraries, ISML 1180 C.9 Runge-Kutta Method 1183 C. 9.1 Background 1183 C.9.2 Description of Method 1183 C. 9.2.1 Illustrative Use of Routines 1188 CIO Runge-Kutta Using C 1191 C.l 1 Runge-Kutta-Fehlberg Method 1193 Index 1207