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
U.S. GEOLOGICAL SURVEY
Water-Resources Investigations Report 96-4240
Table of Contents
Abstract
-
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
-
4.3.1.1
Area
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
Dynamic Characteristics
-
4.3.2.1
Conveyance
4.3.2.2
Flux Coefficients
4.3.2.3
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
-
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
Two-Node Control Structures: Codes 3 and 5
-
8.1.2.1.2.1
Same Elevation: Code 3
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
Three-Node Control Structures
-
8.1.2.1.3.1
Average Elevation at Two Nodes: Code 12
8.1.2.1.3.2
Flow Over a Side Weir: Code 14
8.1.2.2
Variable Geometry
-
8.1.2.2.1
One-Node Control Structures
-
8.1.2.2.1.1
Opening Fraction Given Beforehand: Code 4, Type 4
8.1.2.2.1.2
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
Two-Node Control Structures
-
8.1.2.2.2.1
Explicit Two-Dimensional Flow Tables: Code 5, 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
Operation Control Blocks
-
8.1.2.2.3.1
Gate Control
8.1.2.2.3.2
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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