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
An important example from open-channel flow analysis is the top-width function for a cross section. The top width for a cross section is the width of the water surface at any elevation in the cross section from the minimum point to some user-established maximum point (the elevation domain). For an elevation in the elevation domain, a single value of top width will be determined from the top-width function. Other functions associated with a cross section include the area function, conveyance function, and the wetted-perimeter function. Characteristics of the cross section are viewed as a function because they are the features of a cross section considered in the governing equations.
Other functions of interest in flow analysis include stage-discharge relations at gaging stations, head-discharge relations for a wide variety of special features, elevation-area-storage relations for reservoirs, and inflow hydrographs. Defining these functions is one of the major tasks in the analysis of any stream. Hundreds of functions may have to be defined for even a small to medium-sized stream system.
Most of these functions of interest in flow analysis are not known as simple mathematical expressions, so solution of the governing equations requires description of various functions in a way that is both flexible and convenient. Function tables are used in FEQ simulation for nearly all functions needed in unsteady-flow analysis. A function table consists of a set of selected argument values (the tabulated argument set) and the corresponding set of function values, as well as a rule for defining the function values for arguments not in the tabulated argument set. This approach is taken because most functions of interest are known only approximately, and some error can be allowed in the function value and in the rule used to compute the values not found in the table. Consequently, the characteristics of the cross sections used in FEQ simulation are computed in the utility program FEQUTL (D. D. Franz and C. S. Melching, 1997) and placed in specially designed function tables called cross-section tables. The cross section is not used in simulation except as reflected in the cross-section function table. (The need to store cross-section characteristics in function tables is another major difference between steady-flow and unsteady-flow analysis; the characteristics of the cross section are computed as needed from fundamental (raw) cross-section data in many steady-flow programs.)
The cross section is normally defined by a set of selected points on the periphery of the cross section in some convenient coordinate system; the points are measured in the field or taken from topographic maps with the assumption that adjacent points may be connected with straight lines. The cross section may be subdivided by vertical, frictionless, fictional walls to account for problems with application of the hydraulic radius to describe the shape of the cross section when computing the conveyance for compound and composite channels. A compound channel is a channel whose cross section consists of subsections of variously defined geometric shapes (Yen, 1992, p. 64). The most common example of a compound channel is one with flood plains. A composite channel is a channel whose wall roughness changes along the wetted perimeter of the cross section (Yen, 1992, p. 60). For compound and composite channels, each subdivision also may be assigned a separate value of Manning's n in FEQ simulation to account for variations in roughness along the periphery of the cross section.
The approach of computing the cross-sectional characteristics as required from the fundamental or raw cross-section data is not efficient for unsteady-flow analysis. In steady-flow analysis, cross-sectional characteristic values need be computed only a few times. In unsteady-flow analysis, however, values of cross-sectional characteristics may be needed many thousands of times; therefore, it is economical in terms of computer time to place the computed cross-sectional characteristics in a cross-section function table for later access.
Many types of function tables are supported in the FEQ model. Three broad classes of function tables are one-dimensional, 1-D (one argument, perhaps several functions), two-dimensional, 2-D (two arguments, several functions), and three-dimensional, 3-D (three arguments, several functions). Six options are available for cross-sectional characteristics (1-D table with several functions), three for 2-D tables, four for functions of time (1-D table with one function) such as hydrographs, and three for other 1-D tables. Details on the arguments, the values tabulated, and the methods applied for interpolation are given in Franz and Melching, 1997).
Two styles for the treatment of flow-path end nodes are available in FEQ because of changes and enhancements to the program. In the older style, derived from earlier versions of the program, the flow-path end nodes must be numbered in the range 1 to 1998. The numbers for flow-path end nodes on branches are limited to the range of 1 to 999. In contrast, nodes in the new style must be labeled with alphanumeric information to make stream-system modeling easier. In the new style (figure 3 and figure 4) the upstream flow-path end node on a branch must be composed of the letter "U" followed by the branch number with no intervening spaces. For example, the upstream flow-path end node on branch 51 would be labeled by U51, and the downstream flow-path end node on that branch would be labeled D51. The flow-path end-node label gives both the branch number and the location on the branch. Under the new style, the branch numbers must be in the range of 1 to 999.
The flow-path end nodes on level-pool reservoirs and dummy branches are composed of the letter "F" followed by a number in the new style (figure 3, figure 4, and figure 5). The letter "F" signifies that the node is free of a branch. The number must be in the range of 1 to 999. Consistent use of either odd or even numbers for downstream nodes on these flow paths is preferred. If all the downstream nodes are even and all the corresponding upstream nodes are the preceding odd number, more information about the location and function of each node is conveyed, and modeling is easier and less prone to errors.
A precise and simple sign convention is applied in FEQ for indicating the direction of flow at a node. If the flow is printed as a positive number, it is moving downstream. If the flow is printed as a negative number, it is moving upstream. This also indicates that a positive flow value at an upstream flow-path end node is flow into the path to which the node is attached. For a downstream flow-path end node, a positive flow is flow leaving the path to which the node is attached. A negative flow gives the opposite direction at a flow-path end node.