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
One of the first steps in steady-flow analysis is to locate control points; that is, points along the stream where the elevation can be computed once the steady flow is selected. At least one point of known elevation is needed to start the computations. For subcritical flow, this point, called an initial condition, will be the downstream boundary of the region of interest.
The algebraic governing equation for steady flow does not apply to rapidly varied flow at bridges, culverts, falls, rapids, dams, and other special features. Furthermore, the governing equation does not apply to junctions of two or more channels or to abrupt changes in channel size or shape. These special features must be isolated and analyzed with equations other than those applied for each computational element. Each special feature forms internal control points in the stream system, one upstream and one downstream. Therefore, each stream segment between the boundary and a special feature or between special features is simulated by a separate steady-flow analysis that requires an initial condition. The necessary initial conditions can be computed with the equations relating flow and elevation from upstream to downstream for the special features.
During the first run of a steady-flow analysis, one or more of the computational elements may prove to be too long, and computational failure results. The only recourse is to subdivide the computational element into two or more shorter computational elements and rerun the analysis.
Also during the solution process, an elevation may result such that the flow is supercritical at the upstream end of the computational element when the elevation at the downstream end is for a subcritical flow. This result is incorrect. In steady flow, such a pattern indicates a hydraulic jump somewhere in the computational element. The analyst may consider three possibilities. First, the incorrect solution may be purely a computational artifact resulting from the failure of the solution process to determine the subcritical solution at the upstream end of the computational element. If so, the solution process should be changed to seek a subcritical solution. Second, there may be no subcritical solution for the unknown in the computational element. This also can be a computational artifact if the computational element is too long and the errors in the approximation of the differential or integral terms are distorting the solution. If so, the computational element must be subdivided and the solution tried again. Third, the flow is physically supercritical near the computational element, yielding the incorrect solution. If so, the control point for the supercritical flow must be found, the profile downstream from the supercriticalflow control point must be computed, and the hydraulic jump can then be located by iterating between the downstream propagation of supercritical flow and the upstream propagation of subcritical flow.