Full Equations (FEQ) Model for the Solution of
the Full, Dynamic Equations of Motion for
One-Dimensional Unsteady Flow in Open
Channels and Through Control Structures

By DELBERT D. FRANZ, Linsley, Kraeger Associates and
CHARLES S. MELCHING, U.S. Geological Survey


Water-Resources Investigations Report 96-4240

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Table of Contents

1. Introduction
1.1 Purpose and Scope

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. Relation Between Analyses of Steady and Unsteady Flows
2.1 Channel Segmentation

2.2 Review of Steady-Flow Analysis
2.3 Basic Principles of Unsteady-Flow Analysis
3. Schematization of the Stream Network
3.1 Physical Features
3.1.1 Branches

3.1.2 Dummy Branches
3.1.3 Level-Pool Reservoirs
3.1.4 Special Features
3.2 Computational Simplifications for Schematic Development
3.2.1 Functions and Function Tables

3.2.2 Flow-Path End Nodes
3.2.3 Flow-Sign Convention
3.3 A Physical Analogy of the Schematization of the Stream Network
4. Description of the Channel Geometry
4.1 The One-Dimensional Assumption

4.2 Directional Changes
4.2.1 Channel Geometry Requirements for One-Dimensional Analysis

4.2.2 Orientation of Cross Sections
4.2.3 Selection of Distance Axis
4.3 Characteristics of a Cross Section
4.3.1 Static Characteristics Area First Moment of Area With Respect to the Water Surface Wetted Perimeter Derivatives of Area and First Moment of Area
4.3.2 Dynamic Characteristics Conveyance Flux Coefficients Critical Flow and Critical Depth
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

5.1.2 Conservation of Momentum
5.2 Differential Form of the Equations
5.2.1 The Conservation Form

5.2.2 The Saint-Venant Form
5.2.3 The Characteristic Form
5.3 Nature of Shallow-Water Waves
5.4 Integral Form for Curvilinear Alignment
5.5 Special Terms in the Equations of Motion
5.5.1 Drag for a Submerged Body

5.5.2 Transformation of Energy-Head Losses to Drags for Control Structures
5.5.3 Sudden Cross-Sectional Expansions or Contractions
5.6 Extended Motion Equation
6. Approximation of the Full Equations of Motion in a Branch
6.1 Methods of Mathematical Approximation

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. Equations of Motion for Dummy Branches and Level-Pool Reservoirs
7.1 Approximation of the Equations of Motion in a Dummy Branch

7.2 Approximation of the Equations of Motion in a Level-Pool Reservoir
8. Boundary and Initial Conditions
8.1 Internal Boundary Conditions
8.1.1 Conservation of Mass: Code 2

8.1.2 Elevation-Flow Relations Fixed Geometry One-Node Control Structures: Code 4, Types 1-3; Code 8 Two-Node Control Structures: Codes 3 and 5 Same Elevation: Code 3 Flow Expansion: Code 5, Type 1 Bi-Directional Flow with Pump or Simple Conveyance: Code 5, Type 2 Abrupt Expansion with Lateral Inflow/Outflow: Code 5, Type 5 Explicit Two-Dimensional Flow Tables: Code 5, Type 6 Conservation of Momentum/Constant Elevation: Code 11 Conservation of Momentum/Energy: Code 13 Three-Node Control Structures Average Elevation at Two Nodes: Code 12 Flow Over a Side Weir: Code 14 Variable Geometry One-Node Control Structures Opening Fraction Given Beforehand: Code 4, Type 4 Opening Fraction Computed in Full Equations Model: Code 4, Type 5 Single Pump with Rate Limited by Tail Water: Code 4, Type 6 Two-Node Control Structures Explicit Two-Dimensional Flow Tables: Code 5, Type 6 Variable-Height Weir: Code 5, Type 7 Sluice Gates at Stratton Dam at McHenry, Ill.: Code 5, Type 8 Underflow Gates: Code 5, Type 9 Variable-Head Variable-Speed Pump: Code 5, Type 3 Operation Control Blocks Gate Control Pump Control
8.2 External Boundary Conditions
8.2.1 Flow as Function of Water-Surface Elevation: Code 4 and Code 8

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
8.3 Initial Conditions
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

9.1.2 Application to Simultaneous Equations
9.2 Solution of a Sparse, Banded Matrix
9.3 Stopping Criteria for Newton's Method
9.4 Stability, Convergence, and Accuracy of the Solution Scheme
9.4.1 Truncation Errors

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. Matrix Form of the System of Equations Describing the Network of Channels
10.1 Example: Single Branch

10.2 Example: Two Branches with Overflow Dam
10.3 Example: Three-Branch Junction
10.4 Example: Four Branches with Loop
11. Function Table Description
11.1 One-Dimensional Function Tables
11.1.1 Types 2 and 7

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 Two-Dimensional Function Tables
11.2.1 Type 13

11.2.2 Type 14
11.3 Three-Dimensional Function Tables
11.4 Summary of Function Tables
12. Program Outline
13. Input Description for the Full Equations Model: Version 8.0
13.1 Run Control Block--Run Control Table

13.2 Branch Description Block--Branch Tables
13.2.1 Computational-Element Interpolation

13.2.2 Storm Sewers as Branches
13.3 Tributary Area Block--Tributary Area Tables
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
References Cited

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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