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
The first step in applying FEQ to the analysis of a stream system is the development of a schematic diagram that subdivides the stream system into a series of connected flow paths. A flow path conveys water between points in the stream system. Examples of flow paths, as applied in FEQ, are a stream channel, canal, storm sewer, or reservoir. Additional examples of flow paths on the stream are an overflow spillway, a swale that carries water overland during floods, or a breach in a levee or dam. In FEQ, these flow paths are connected by special features, which include culverts, bridges, dams, junctions, sluice gates, and other components of the stream system that do not fit the concept of flow paths considered in FEQ (branches, dummy branches, and level-pool reservoirs). The conceptual descriptions of each of the flow paths are given in sections 3.1.1-3.1.3. The primary hydraulic characteristic that distinguishes special features from flow paths in FEQ simulation is that the special features are potentially major flow transitions, either natural or constructed, small enough that changes in storage and momentum content can be neglected and relations between water-surface elevation and discharge can be derived from steady-flow principles. More than 20 special features are considered in FEQ. Thus, the stream system for unsteady-flow analysis with FEQ can be described in a schematic diagram that shows the branches, dummy branches, and level-pool reservoirs and the connections with the special features. The flow paths and special features are discussed below and in subsequent sections.
To draw the schematic diagram, the locations and types of special features in the stream system must be identified. Once the schematic diagram has been drawn, the flow paths must be labeled. The end nodes of a flow path have a special status because they connect the flow path to the rest of the stream system. A flow-path end node defines the end of the flow path. Each flow-path end node is labeled. A labeled schematic diagram is the basis for describing the various connections of the stream system as modeled in FEQ. The schematic diagram also defines how all parts of the stream system are to be modeled with the mathematical relations available in FEQ and the companion utility program, Full EQuations UTiLities (FEQUTL) (D. D. Franz and C. S. Melching, 1997).
A map of an example stream system is shown in figure 2, and a sample schematic for this stream system is shown in figure 3. Each special feature has been isolated and the flow paths begin and end with a flow-path end node. Some of the flow-path end nodes on branches have been labeled in figure 3. The rule for labeling is very simple: The upstream flow-path end node on a branch is denoted by the letter "U" followed by the branch number, and the downstream flow-path end node is denoted by the letter "D" followed by the branch number. Because the rule is simple and is tied to the branch number, labels for flow-path end nodes are commonly omitted on branches in the FEQ schematic. The nodes on a level-pool-reservoir flow path or on a dummy-branch flow path must be labeled. The label is formed by the letter "F" followed by a number chosen by the user, as defined below. The stream system shown in figure 2 includes two run of the river dams that are represented by branches in FEQ simulation, as illustrated in figure 3.