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 various forms of governing equations progress from physical to mathematical forms. The integral form and the conservation form relate closely to the fluxes and forces acting on the flow. In contrast, the relation to the fluxes and forces is missing in the Saint-Venant and characteristic forms. The characteristic form has lost almost all reference to the forces and fluxes included in the equation for the conservation of momentum; however, the characteristic form provides insight into shallow-water wave motion, which is not evident in the other forms. This insight is vital to understanding the requirements (boundary and initial conditions) that must be met as approximate solutions to these equations are sought. Thus, no single form of the governing equations is adequate for understanding unsteady 1-D open-channel flow.
The insight to be gained from the characteristic form is best visualized by tracking small but identifiable disturbances in a stream channel. Consider a long rectangular stream channel with no special features, a branch in the FEQ network schematization. Also, imagine that the flow is steady and subcritical but nonuniform. To introduce a small shallow-water-wave disturbance, a short segment of the channel bottom is made of some flexible material that can be given a sudden sharp but small upward displacement. This displacement disturbs the whole column of water above the location of the flexible strip and is analogous to the mechanism thought to initiate tsunamis in the Pacific Ocean. Because the flow is subcritical, a shallow-water wave will move upstream and downstream. To track each of these small waves, the location of the waves along the channel is measured periodically. The path or trajectory of the wave or waves can then be depicted by use of a coordinate system in which the distance along the channel, x, is shown on the horizontal axis and the time, t, is shown on the vertical axis, as in figure 10. This coordinate system defines the x-t plane.
Suppose that the disturbance was introduced at station XL at time t = t0. Small shallow-water waves will travel upstream and downstream from this station. The upstream wave will have a velocity V - c and the downstream wave a velocity of V + c. The trajectory of the upstream wave is denoted as C- and of the downstream wave is denoted as C+ in figure 10. The region of the x-t plane between these two trajectories is the region of influence of the disturbance at point XL at time t = t0. Outside this region, the disturbance has no effect on the flow. Another disturbance has been introduced at a station XR, some distance downstream from XL. The C+ and C- trajectories for this disturbance also are shown on figure 10. The region of the x-t plane between the C+ trajectory of the disturbance at XL and the C- trajectory of the disturbance at XR is called the domain of uniqueness because the flows in this region cannot be affected by disturbances upstream from XL or downstream from XR originating at any time t greater than or equal to t0. The distance interval from XL to XR is called the interval of dependence for point G because the flow at point G is dependent on knowledge of the flow on this interval at time t = t0.
These features of shallow-water-wave motion are important in designing methods to compute approximations to the motion. In these methods, a known condition in the stream channel at a time t = t0 is applied, and the conditions in the channel at some later time are computed with the equations of motion. The known conditions at which the computations start are called initial conditions. For practical reasons, the initial flow condition is almost always assumed to be a steady flow. To adequately estimate conditions at point G from information at t = t0, information must be available about the flow on the interval of dependence for point G. If that information is not available, meaningful estimation cannot be made because unknown conditions have the potential to affect the values at point G. Thus, to estimate the conditions at any point on the x-t plane, information must be available about the interval of dependence for that point.
This requirement for information on the interval of dependence for each point in the channel has implications at the boundaries of the channel. Every channel is of finite length; at some point the analysis starts and at another it ends, so boundaries must be defined. Possible conditions at the boundaries of a channel are shown in figure 11. If the flow is subcritical, the interval of dependence for the upstreammost point on the channel is somewhat upstream from the boundary point. Thus, estimation of flow conditions at this boundary point requires information about the flow conditions upstream of the boundary. The C+ trajectory from upstream points affects the flow at that point in the x-t plane, yet the analysis for this channel stops at the boundary. Therefore, a single condition must be specified at the boundary point. This condition, called a boundary condition, may be one of three types: flow as a function of time, water-surface elevation as a function of time, or a relation between flow and water-surface elevation. This condition supplies the information that is lacking along the upstream boundary for the channel. The same applies at a downstream boundary if the flow is subcritical. Again, part of the interval of dependence falls outside the channel length being analyzed, and a downstream-boundary condition must be supplied.
If the flow is supercritical at a boundary, the required number of conditions changes. At an upstream boundary, the flow and the elevation of water must both be supplied because, as shown in figure 11, the interval of dependence of points on the upstream boundary in the x-t plane are outside the length of channel analyzed. At the downstream end, no boundary conditions are required because the interval of dependence falls within the length of channel analyzed. Thus, the number of boundary conditions required depends on whether the flow is subcritical or supercritical at the boundary.