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Full Equations Utilities (FEQUTL) Model for the Approximation of Hydraulic Characteristics of Open Channels and Control Structures During Unsteady Flow


4.1 Bridges

The Water-Surface Profile Computations (WSPRO) computer model described in Shearman (1990) and Shearman and others (1986) is applied in FEQUTL to compute a description of the hydraulics of bridges. A set of commands is provided in FEQUTL to assist the user in creating a 2-D table of type 14 in which the head upstream from the bridge is a function of the downstream head and partial free flow through the bridge for application in FEQ simulation. Head is the water-surface elevation less the elevation of the datum for the bridge. This approach was taken to avoid the extensive effort required to include the WSPRO methods in FEQUTL. WSPRO, FEQUTL, and FEQ must be applied and preferably well known by the user to simulate flow through bridges. WSPRO software is widely distributed and is in use by several organizations.

WSPRO is applied to compute multiple water-surface profiles through the structure to define the range of flows and downstream and upstream heads expected during unsteady-flow simulation. This means that 100 to 400 profiles may be required to properly define the table of type 14. Three commands are provided in FEQUTL to make the process of defining these profiles easier.

Application of the WSPROX command (section 5.23) results in extraction of cross-section data from a WSPRO input file and reorganization of these data into FEQXEXT format (section 5.9) for later computation in FEQUTL. The user also can request that the cross-section tables be computed in FEQUTL directly without the intermediate step of placement in the FEQXEXT format. The WSPROX command is used to extract cross sections from available WSPRO input. The cross sections of interest are those required in both WSPRO and FEQ to represent the hydraulics of bridges. The approach and exit sections for the bridge are required in both WSPRO and FEQ. The approach section is usually one bridge-opening width upstream from the bridge. If spur dikes are present, the approach section is one spur-dike opening width upstream from the opening of the spur dikes. The exit section from the bridge is about one bridge-opening width downstream from the bridge and not at the downstream face of the bridge. Shearman (1990) provided details on the selection of the approach and exit sections. The approach section will be at the downstream end of the branch upstream from the bridge and the exit section will be at the upstream end of the branch downstream from the bridge in the stream-network schematization applied in FEQ. In general applications of FEQUTL, WSPROX may be used to convert all cross-section data from a WSPRO input file into function tables for use in FEQ simulation.

The cross sections between the approach and exit section required in WSPRO are not used in the FEQ model. The effects of those cross sections are implicit in the type 14 function table describing the hydraulics of the bridge. This table must describe the relations among three quantities: the water-surface elevation in the approach section, the water-surface elevation in the exit section, and the flow between these two cross sections. Flow through bridge openings, flow through culverts, and flow over the roadway are calculated in WSPRO. Any of these flow paths can appear in the WSPRO description of the structure. The resulting function table will list the information required in FEQ to simulate the hydraulics between the approach and exit sections of the bridge.

WSPRO is applied to compute the water-surface elevation in the approach section given a series of flows and water-surface elevations in the exit section. These water-surface elevations and flows are placed in a 2-D table of type 14. The water-surface elevation at the approach section for a range of flows must be computed in WSPRO, for each water-surface elevation in the exit section. The flow range must include all flows expected for each exit-section water-surface elevation. The range of flows that must be defined depends on the nature of the flows at the bridge. For example, if the bridge is not subject to backwater effects, the range of flows can be narrow, only including the possible range of flows at a given elevation resulting from variations in water-surface slope as a flood wave passes. Conversely, if the bridge is subject to substantial backwater effects, such as resulting from a gated spillway on a dam, the range of flows must be large. The maximum flow of the range of flows is treated in FEQ simulation as a free flow (free of backwater effects). This will not be true in general because the free-flow limit for the structure cannot be defined in WSPRO computations. This means that the maximum flow in the table for each exit-section water-surface elevation must be larger than any flow that may be computed for that elevation during the unsteady-flow simulation. If a larger flow is computed during simulation than is in the table, flow through the bridge will be simulated as free flow and the water-surface elevations in the table describing the hydraulics of the bridge will be erroneous. The range of flows needed for each exit-section water-surface elevation must be established by the user with careful consideration of these requirements. The user must carefully check the computed downstream heads and flow rates at bridges to ensure that the maximum flow tabulated for each exit-section water-surface elevation was not exceeded.

The WSPROQZ command (section 5.21) can be used for setting the range of flows for a series of exit-section (downstream) water-surface elevations. The user provides the series of downstream water-surface elevations as well as information that defines the maximum flow rate for each elevation. This information can be a flow value, a normal-depth rating computed using the exit cross section and a user-supplied friction slope, or a critical-flow cross section. The user also supplies the smallest partial free flow to compute in the table. By selecting a friction slope larger than is possible for the bridge location, the user can force the maximum flow to be large. However, if the maximum flow is too large, the flow in the bridge opening will become supercritical and the computations in WSPRO may fail. The bridge-opening cross section can be given as the critical-flow cross section, at least when the opening is flowing part full, to limit the maximum flow defined by a friction slope. Additional details are provided in section 5.21.

The flow and water-surface-elevation input lines and WSPRO comment and output-specification lines are prepared when the WSPROQZ command is applied. All other WSPRO information, such as cross-section geometry and flow-resistance and head-loss coefficients, must be prepared by the user in accordance with the formats given in Sherman (1990). The WSPRO input lines prepared in WSPROQZ are then transferred by the user to the manually prepared input file for the WSPRO description of the bridge. A series of water-surface profiles is computed with WSPRO as specified in the input lines prepared in WSPROQZ. The output-specification lines prepared in WSPROQZ result in a WSPRO output format that may be converted to a table of type 14 in FEQUTL. More than one run of WSPRO may be required because of profile storage limitations of the particular executable version of WSPRO.

The printer-output file or files computed with WSPRO are accessed and a 2-D function table is prepared with the WSPROT14 command (section 5.22). The partial free flows from a value of 1.0 to the minimum given by the user plus a value at a partial free flow of 0.0 are placed in the table. The user must ensure that the flow range is adequate and that linear interpolation between tabulated values is acceptably accurate.

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