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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

U.S. GEOLOGICAL SURVEY WATER-RESOURCES INVESTIGATIONS REPORT 96-4240

3.1 Physical Features

The physical (geomorphologic) features of the stream system are divided into four categories in the stream-network schematization applied in FEQ: branches, dummy branches, level-pool reservoirs, and special features. Detailed descriptions and examples of each of these categories of physical features and the methods used to characterize these features in FEQ are given in the following sections.

3.1.1 Branches

A branch is the length of channel between special features (boundaries, junctions, and flow-control structures). Flow through a branch is described by the governing equations described in section 5, and in this sense every branch is identical. A branch is subdivided into computational elements for developing the approximate algebraic governing equations (see section 6). A branch also has nodes at the boundaries of these computational elements, each node representing an associated cross section. The nodes at the two ends of the branch are called flow-path end nodes, and those not on the ends are called interior nodes. The branch has an upstream end and a downstream end that the user must assign. For example, the node at the upstream end is called the upstream flow-path end node. The nodes on a branch are numbered for identification, the numbers increasing and consecutive from the upstream end to the downstream end. A station--the distance measured along the stream from some convenient reference point--is assigned to each node on the branch. The elevation of the minimum point in the cross section is a key feature of each node on a branch. The station-elevation pair for each node defines the bottom profile of the stream channel. The absolute value of the difference in the stations of two consecutive nodes on a branch gives the length of the computational element between the two nodes.

Water can enter a branch in three ways: as inflow at the two ends or inflow from the area tributary to the branch. Thus, an associated tributary area may be assigned to each branch and computational element. The tributary area for a computational element is the area that will contribute lateral inflow to the computational element. Lateral inflow to a computational element may include diffuse overland flow, seepage into (or out of) the channel, and point discharges from tributary streams or storm sewers too small to simulate explicitly. Most often the lateral inflow from a tributary area is estimated by a hydrologic model producing unit-area values of runoff intensity for one or more types of land cover. For example, the area tributary to a computational element may consist of agricultural, forest, and urban land. Each of these land covers would have a different rainfall-runoff relation in the hydrologic model. Therefore, it is convenient to allow the subdivision of the tributary area into the different land-cover types used in the hydrologic model.

An additional factor to the estimation of lateral inflow into the computational element is the gage; that is, the precipitation gage where rainfall was measured from which the runoff was computed. More than one gage may be available in a watershed. To consider multiple gages in a watershed in FEQ simulations, the tributary area for each computational element must be associated with the gage used to compute the unit-area runoff intensity.

A final feature of a branch required in FEQ simulation is a way of identifying and referencing the branch. Each branch defined in FEQ must be given a positive number for this purpose. Branch numbers can be nonconsecutive, but they have an upper limit as discussed below in the section on Flow-Path End Nodes (3.2.2).

3.1.2 Dummy Branches

A branch conveys water along a certain flow path whose characteristics include depth, area, and length. Other flows paths have these characteristics, but these details are not of interest in FEQ application. An example is the flow of water over the emergency spillway of a reservoir: depths in the spillway and the associated channel are of interest when these items are designed, but only flow over the emergency spillway is of primary interest in an unsteady-flow analysis. The water volume in the short, steep discharge channel associated with the spillway is too small to have an effect on the results, so a dummy branch is designed to represent such a flow path, as illustrated in figure 4.

A dummy branch has two flow-path end nodes but no associated cross sections. The only values of interest are the water-surface elevations and the flows at the nodes. For computational purposes, a small storage and friction loss must be assigned to the dummy branch, but these values are set so small that the flows and elevations at the two nodes are nearly equal. Other examples of the application of dummy branches include flows over a levee, multiple outflow paths through or around a dam, and intermittent flow of water over land connecting two streams or reservoirs.

3.1.3 Level-Pool Reservoirs

The final flow path as previously defined is a level-pool reservoir. Storage volume of a level-pool reservoir is large enough relative to the volume of flow entering and leaving the reservoir that the water surface can be treated as horizontal with only a small error in the results. A level-pool reservoir, like a branch and a dummy branch, has two flow-path end nodes (fig. 5). One node represents inflow to the reservoir, and the other node represents outflow. Long and narrow reservoirs or lakes often do not conform to the level-pool assumption and should be treated as branches in FEQ simulation because the flows result in an appreciable slope on the water surface.

3.1.4 Special Features

The identification and description of special features in a stream system is a major part of building a mathematical model of the stream system. A key hydraulic aspect of special features is their size; these features are so small that storage and momentum content changes may be neglected and relations between water-surface elevation and discharge may be derived from steady-flow principles. The variety of special features in streams systems is endless, especially in urban streams. The following are some examples of special features:

  1. Junctions between or among tributaries or distributaries. Junctions are locations where two or more channels meet and combine to form a single channel (figure 1 and figure 2). Locations where a single channel splits and forms two or more channels also are junctions. Multiple inflows for the inflow node of a level-pool reservoir also can be represented by a junction (fig. 5). Junctions are always present at connections between flow paths; they establish the relation among the flows in the flow paths at the connections.

  2. Points of known water-surface elevation or of known or knowable flow. These points are generally the logical places for boundary conditions. The values of water-surface elevation or flow can be functions of time and need not be constants.

  3. Points of known relations between water-surface elevation and flow rate, such as streamflow-gaging stations. These also make good boundary conditions especially at the downstream boundary.

  4. Any change in bottom slope that might be large enough to result in a critical control. Critical controls must be isolated because branches that include supercritical flow must be treated differently than those characterized entirely by subcritical flow. Locations of potential critical flow must be isolated for proper analysis of steady or unsteady flow.

  5. Any abrupt change in channel shape or roughness. These transitions must be isolated to account for the additional expansion or contraction losses.

  6. Dams, control weirs, and large pumping stations. These and similar features can have a substantial effect on the water-surface elevation or the amount of water flowing in the stream.

  7. Drop structures, falls, or rapids. These will be control points, at least at certain flow levels. These control points can be drowned out at high flows and reestablished at lower flows in FEQ simulation.

  8. Bridges and culverts that are to be represented explicitly.

  9. Points at which a special feature may be added to improve the control of the stream system.

Anything that is not a branch, a dummy branch, or a level-pool reservoir is a special feature. The list of special features can be further grouped into the general classes of junctions, boundary conditions, and control structures.

Control structures can be further grouped into several subclasses. A control structure is any physical feature that exerts a measure of control on the flow. If the control of flow is complete, so that a unique relation between flow and water-surface elevation is established by the structure, then the structure is called a one-node control structure because only the value of flow or elevation need be known at one flow-path end node to fully define the other value. If the control is incomplete, in that knowledge of the water-surface elevation at two flow-path end nodes is needed before the flow is defined, then the structure is called a two-node control structure. A major challenge of unsteady-flow analysis is often the identification and description of the control structures.

3.1.1 Branches
3.1.2 Dummy Branches
3.1.3 Level-Pool Reservoirs
3.1.4 Special Features
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