OPEN CHANNEL FLOW. Computer Applications. Numerical Methods and. Roland Jeppson. CRC Press UNIVERSITATSB'BUOTHEK TECHNISCHE. INFORMATlONSBiBUOTHEK

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1 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

2 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 Introduction One-, Two-, and Three-Dimensional Flows Steady versus Unsteady Flow Uniform versus Nonuniform Flow Prismatic versus Nonprismatic Channels Subcritical, Critical, and Supercritical Flows Turbulent versus Laminar Flow Review of Basic Fluid Mechanics Principles Physical Properties of Fluids and Their Effects on Open-Channel Flows Conservation of Mass, or Continuity Equations Energy Principle Kinetic Energy Correction Coefficient, a Momentum Principle in Fluid Flow Momentum Flux Correction Coefficient, 3 34 Problems 38 Chapter 2 Energy and Its Dissipation in Open Channels Introduction Approaches to Frictional Resistance Friction Factors in Open Channels Combining the Chezy and the Chezy C Equations Empirical Formula: Use of Manning's Equation Channels with Varying Wall Roughness, but = Q Constant Specific Energy, Subcritical and Supercritical Flows Flumes Delivery Diagrams Graphical Aids to Solving Critical Flow Problems Upstream Depth When Critical Conditions Occur at Reduced Downstream Section Dimensionless Treatment of Upstream Trapezoidal Channel to Downstream Rectangular Channel Upstream Channel Also Rectangular Hydraulically Most Efficient Section Nondimensional Variables for This Section 155 Problems 157 References 182

3 vi Chapter 3 The Momentum Principle Applied to Open Channel Flows The Momentum Function Characteristics of the Momentum Function Rectangular Channels and Momentum Function per Unit Width Polynomial Form for Momentum Function Dimensionless Momentum Functions Celerity of Small Amplitude Gravity Waves Constant Height Waves Open Channel to Pipe Flow Multiple Roughness Coefficient for Channel Section Compound Sections 233 Problems 236 Problems to Solve Using Program CHANNEL 253 Chapter 4 Nonuniform Flows Types of Nonuniform Flows Ordinary Differential Equation for Gradually Varied Flow Bulk Lateral Outflow Lateral Inflow Generalization of Gradually Varied Flow Equations Gradually Varied Flow in Prismatic Channels without Lateral Inflow or Outflow Classification of Gradually Varied Profiles Sketching GVF Profiles in Prismatic Channels Alternative Forms of the ODE That Describe GVF Profiles Numerical Methods for Solving ODEs Canal Systems Simultaneous Solution of Algebraic and Ordinary Differential Equations Flow into a Mild Channel with a Downstream Control Different Modes of Gate Operation Hydraulic Jump Downstream from a Gate in a Finite Length Channel Nonprismatic Channels Culverts Solutions When Upstream Control Exists Solutions When Downstream Control Exists GVF Profiles in Nonprismatic Channels GVF Profiles in Branched Channel Systems GVF Profiles in Parallel Channels Solutions to Spatially Varied Flows Outflow from Side Weirs Closed-Form Solution to Side Weir Outflow Spatially Varied Inflows Algebraic Solution Spatially Varied Flow in Nonprismatic Channels Tile Drainage Downstream Controls in Nonprismatic Channels 433

4 vii 4.20 Gutter Flow and Outflow through Grates Gutter Flow Flow into Grates at Bottom of Channel Combined Problem: Gutter Inflow and Grate Outflow Lateral Inflow over Grate Length No Negative Flow Rates Last Grate at End of Gutter Subcritical Flow through n Gutter-Grates Multiple Branched Channel Systems Other Dependent Variables in GVF Computations Varied Flow Function Moving Waves Quasi-Unsteady Analysis Downstream Controlled Waves Explicit Method Implicit Method Upstream Controlled Waves Moving Hydraulic Jump 578 Problems 593 References 656 Chapter 5 Common Techniques Used in Practice and Controls Introduction Resistance to Flow in Natural Streams and Rivers Techniques Used for Solving Steady Flows in Irregular Channels Defining Irregular Channel Properties Use of Cubic Splines to Define Cross Section Solving Manning's and Energy Equations in Natural Channels Hand Solution to GVF-Flow by the Standard Step Method Using an ODE Solver to Compute GVF-Profile in Irregular Channels Multiple Parallel Channels and Flood Plain Storage Implementation of Solution for Compound Channel System of Natural Streams and Rivers HEC-2 Water Surface Profiles Water Measurement in Channels Current Meters Weirs Parshall Flumes and Cutthroat Flumes Other Critical Flow Flumes Design of Transitions Subcritical Transitions Design of Transitions by Hand Computations Transition Design by Solving an ODE Problem Supercritical Transitions Channel Contractions Design of Supercritical Transitions Channel Enlargements Gates 741

5 vjjj 5.7 Submerged Flow Downstream from Vertical Gates Dimensionless Forms of Equations Series of Submerged Gates Design of Side Weirs Optimal Design of Trapezoidal Channels Considering Total Costs Additional Optional Analyses 788 Problems 793 References 822 Chapter 6 Unsteady Flows When Should Flow Be Handled as Unsteady? Basic One-Dimensional Equations for Unsteady Channel Flows (The St. Venant Equations) Q-Y Set of St. Venant Equations V-Y Set of St. Venant Equations Derivation Based Directly on Newton's Second Law of Motion No Lateral Inflow or Outflow and Prismatic Channels Changes in Dependent 6.3 Determination of Mathematical Type of St. Venant Equations 828 Variables Second-Order PDEs Taking Advantage of the Equation 6.5 Solution to Unsteady Flows That Deviate Only Slightly Characteristics 831 from Uniform Conditions Boundary Conditions Depth at Origin Specified, Y(0, t) Known Velocity at Origin Specified, V(0, t) Known Flow Rate q at the Origin Specified, q(0, t) Known Upstream Reservoir Water Surface Specified, H(0, t) Known Understanding Characteristics Better Maximum Possible Flow Rates Maximum Point Outflow, Aq Extending the Methods to Nonrectangular Channels Trapezoidal Channels Dimensionless Variables Circular Channels Maximum Flow Rates in Nonrectangular Channels Maximum Outflow AQ That Can Be Taken at an Intermediate Position Positive Waves Control Structures Partial Instant Opening of Gates in Rectangular Channels Partial Instant Closing of Gates in Trapezoidal Channels Partial Instant Closure Followed by Slow Movement Thereafter Rectangular Channels Nonrectangular Channels Dam Break Problem 937 Problems 938

6 i ix Chapter 7 Numerical Solution of the St. Venant Equations Background Method of Characteristics Boundary Conditions Downstream End Upstream End Solution to Boundary Unknowns Using Characteristics with Specified Time Increments Based on Second-Order Approximations Upstream Boundary Condition Equations Downstream Boundary Condition Equations Iterative Solution Technique Explicit Evaluation of Variables at Points L and R Accuracy of Numerical Solutions Implicit Methods Direct Implicit Finite Differencing Boundary Conditions Solving the Difference Equations Computer Code to Implement Direct Implicit Method Alternative Approach to Boundary Condition Equations Gauss-Seidel or Successive-Over-Relaxation (SOR) Iterative Solution Techniques Crank-Nicolson Newton Iterative Implicit Method Weighting Current and Advanced Time Steps Differently The Preissmann Implicit Method Double Sweep Method of Solution Boundary Conditions Downstream Boundary Conditions Upstream Boundary Conditions Solving Preissmann Difference Equations Using the Newton Method 1075 i 7.14 Two-Dimensional Free Surface Flows 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 Rectangle 1105 A.l.2.2 Trapezoid 1106 A Circle 1107 A.l.2.4 Problems (For You to Work) Obtaining Geometric Properties of Irregular Cross Sections 1114 U21

7 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 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 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 Illustrative Use of Routines 1188 CIO Runge-Kutta Using C 1191 C.l 1 Runge-Kutta-Fehlberg Method 1193 Index 1207