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 initial value of flow and water-surface elevation must be known at every node in the stream model before the unsteady-flow computations can begin. These initial values are provided through a steady-flow water-surface profile computation. A steady-flow analog of the unsteady-flow governing equations for branches is used in the steady-flow water-surface profile computations. Most control structures are not represented in these computations. An estimate of the initial conditions is obtained from the steady-flow computations so that the unsteady-flow computations can start. Consequently, an option is provided for holding the boundary conditions and the simulation time constant while initial values of water-surface height are computed. This option is called frozen time. If the frozen-time option is selected by the user, the changes in water-surface height and flow that would take place over a time step are computed, but then the simulation time is reset to the starting time. Normally, only a few frozen time steps must be computed to dissipate transient conditions resulting from the change from steady flow to unsteady flow. If the frozen-time option is selected, a maximum of nine frozen time steps will be computed. In most cases, nine frozen time steps will suffice to reduce computational transitions; however, for simulations of tidally affected flows or streams with many control structures, the computational transients can be particularly strong. In these cases, boundary conditions may have to be held constant (using a hypothetical period of constant conditions) during the start of the unsteady-flow computations (after frozen time) for a period long enough to dissipate the transients induced by the approximate initial condition.
The equations for the steady-flow analysis are not presented here because they are a special case of the governing equations for unsteady flow. A subcritical solution to the governing equations is sought, but (as outlined in section 2.2) a subcritical solution may not exist. If a subcritical solution cannot be computed, then the iterative solution procedure will fail and computations will stop in FEQ. The user must then determine why supercritical flow results. Is the distance step too long or is the slope too steep for subcritical flow? Details of this determination are given in section 13.