the Full, Dynamic Equations of Motion for

One-Dimensional Unsteady Flow in Open

Channels and Through Control Structures

CHARLES S. MELCHING, U.S. Geological Survey

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- 1. Introduction
- 1.1 Purpose and Scope
- 2. Relation Between Analyses of Steady and Unsteady Flows
- 2.1 Channel Segmentation
- 3. Schematization of the Stream Network
- 3.1 Physical Features
- 3.1.1 Branches
- 3.2.1 Functions and Function Tables
- 4. Description of the Channel Geometry
- 4.1 The One-Dimensional Assumption
- 4.2.1 Channel Geometry Requirements for One-Dimensional Analysis
- 4.3.1 Static Characteristics
- 4.3.1.1 Area
- 4.3.2.1 Conveyance
- 5. Full, Dynamic Equations of Motion for One-Dimensional, Unsteady Flow in Open Channels
- 5.1 Integral Form of the Equations
- 5.1.1 Conservation of Mass
- 5.2.1 The Conservation Form
- 5.5.1 Drag for a Submerged Body
- 6. Approximation of the Full Equations of Motion in a Branch
- 6.1 Methods of Mathematical Approximation
- 7. Equations of Motion for Dummy Branches and Level-Pool Reservoirs
- 7.1 Approximation of the Equations of Motion in a Dummy Branch
- 8. Boundary and Initial Conditions
- 8.1 Internal Boundary Conditions
- 8.1.1 Conservation of Mass: Code 2
- 8.1.2.1 Fixed Geometry
- 8.1.2.1.1 One-Node Control Structures: Code 4, Types 1-3; Code 8
- 8.1.2.1.2.1 Same Elevation: Code 3
- 8.1.2.1.3.1 Average Elevation at Two Nodes: Code 12
- 8.1.2.2.1 One-Node Control Structures
- 8.1.2.2.1.1 Opening Fraction Given Beforehand: Code 4, Type 4
- 8.1.2.2.2.1 Explicit Two-Dimensional Flow Tables: Code 5, Type 6
- 8.1.2.2.3.1 Gate Control
- 8.2.1 Flow as Function of Water-Surface Elevation: Code 4 and Code 8
- 9. Matrix Solution and Numerical Properties of the Finite-Difference Scheme
- 9.1 Newton's Iteration Method for Solution of Nonlinear Equations
- 9.1.1 Application to a Single Equation
- 9.4.1 Truncation Errors
- 10. Matrix Form of the System of Equations Describing the Network of Channels
- 10.1 Example: Single Branch
- 11. Function Table Description
- 11.1 One-Dimensional Function Tables
- 11.1.1 Types 2 and 7
- 11.2.1 Type 13
- 12. Program Outline
- 13. Input Description for the Full Equations Model: Version 8.0
- 13.1 Run Control Block--Run Control Table
- 13.2.1 Computational-Element Interpolation

1.2 Classification of One-Dimensional Steady Flow and Unsteady Flow

1.3 Selection of Conservation Principles

1.4 Major Assumptions in Unsteady-Flow Analysis

1.5 Examples of Unsteady-Flow Analysis

2.2 Review of Steady-Flow Analysis

2.3 Basic Principles of Unsteady-Flow Analysis

3.1.2 Dummy Branches

3.1.3 Level-Pool Reservoirs

3.1.4 Special Features

3.2.2 Flow-Path End Nodes

3.2.3 Flow-Sign Convention

4.2 Directional Changes

4.2.2 Orientation of Cross Sections

4.2.3 Selection of Distance Axis

4.3.1.2 First Moment of Area With Respect to the Water Surface

4.3.1.3 Wetted Perimeter

4.3.1.4 Derivatives of Area and First Moment of Area

4.3.2.2 Flux Coefficients

4.3.2.3 Critical Flow and Critical Depth

5.1.2 Conservation of Momentum

5.2.2 The Saint-Venant Form

5.2.3 The Characteristic Form

5.4 Integral Form for Curvilinear Alignment

5.5 Special Terms in the Equations of Motion

5.5.2 Transformation of Energy-Head Losses to Drags for Control Structures

5.5.3 Sudden Cross-Sectional Expansions or Contractions

6.2 Equations of Motion Options

6.3 Rules for Approximate Integration

6.4 Conservation of Mass

6.5 Conservation of Momentum

6.6 Selection of Weights

7.2 Approximation of the Equations of Motion in a Level-Pool Reservoir

8.1.2 Elevation-Flow Relations

8.1.2.1.2 Two-Node Control Structures: Codes 3 and 5

8.1.2.1.2.2 Flow Expansion: Code 5, Type 1

8.1.2.1.2.3 Bi-Directional Flow with Pump or Simple Conveyance: Code 5, Type 2

8.1.2.1.2.4 Abrupt Expansion with Lateral Inflow/Outflow: Code 5, Type 5

8.1.2.1.2.5 Explicit Two-Dimensional Flow Tables: Code 5, Type 6

8.1.2.1.2.6 Conservation of Momentum/Constant Elevation: Code 11

8.1.2.1.2.7 Conservation of Momentum/Energy: Code 13

8.1.2.1.3.2 Flow Over a Side Weir: Code 14

8.1.2.2.1.2 Opening Fraction Computed in Full Equations Model: Code 4, Type 5

8.1.2.2.1.3 Single Pump with Rate Limited by Tail Water: Code 4, Type 6

8.1.2.2.2.2 Variable-Height Weir: Code 5, Type 7

8.1.2.2.2.3 Sluice Gates at Stratton Dam at McHenry, Ill.: Code 5, Type 8

8.1.2.2.2.4 Underflow Gates: Code 5, Type 9

8.1.2.2.2.5 Variable-Head Variable-Speed Pump: Code 5, Type 3

8.1.2.2.3.2 Pump Control

8.2.2 Flow as Function of Time: Code 6, Type 1

8.2.3 Water-Surface Elevation as Function of Time: Code 6, Type 2

9.1.2 Application to Simultaneous Equations

9.3 Stopping Criteria for Newton's Method

9.4 Stability, Convergence, and Accuracy of the Solution Scheme

9.4.2 Verification of the Accuracy and Convergence of the Full Equations Model

9.4.3 Verification of the Full Equations Model on the Fox River, Illinois

10.2 Example: Two Branches with Overflow Dam

10.3 Example: Three-Branch Junction

10.4 Example: Four Branches with Loop

11.1.2 Types 3 and 8

11.1.3 Types 4 and 9

11.1.4 Type 11

11.1.5 Cross-Section Function Tables (Types 20-25)

11.2.2 Type 14

11.4 Summary of Function Tables

13.2 Branch Description Block--Branch Tables

13.2.2 Storm Sewers as Branches

13.4 Branch-Exterior Node Block--Branch-Exterior Node Table

13.5 Near-Zero-Depth Block--Near-Zero Depth Table

13.6 Network-Matrix Control Block--Network-Matrix Table

13.7 Point Flows Block--Point Flow Tables

13.8 Wind Information Block--Wind Table

13.9 Special-Output Locations Block--Special Output Table

13.10 Input Files Block--Input File Table

13.11 Output Files Block--Output File Table

13.12 Operation of Control Structures Block--Operation Tables

13.13 Function Tables Block--Function Tables

13.14 Free-Node Initial Conditions Block--Free-Node Table

13.15 Backwater Analysis Block--Backwater Table

Appendix
1: List of Notation

Appendix
2: Unformatted Data-File Structures: Full Equations Model Version 8.0

Appendix
3: Dimensions of Arrays and Specification of Fortran Unit Numbers in the
Full Equations Model

Appendix
4: Full Equations Model Error Messages, Warnings, and Bugs: Version 8.0

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