,
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
Cunge and others (1980, p. 36) note that flow as a function of water-surface elevation cannot be used as an external upstream boundary, because the flow would increase without bound. This results because an increase in flow at an external upstream boundary will cause an increase in stage, and this increase in stage will cause an increase in flow from the one-node control-structure relation. Therefore, this boundary condition can only be used as an external downstream boundary.
(121)
,
where DD =1 if positive values from f qb enter the stream system at this external boundary node and DD= -1 if positive values from fqb leave the stream system at this external boundary node. This specification of flow at the node will be met for all conditions in the modeled stream system.
Improper use of a boundary of this type can result in gross errors. Typically, flow as a function of time is given at the upstream (inflow) node of a flow path as an external boundary. This implies that downstream conditions do not affect flow at that node. If an effect from downstream is evident, then the boundary point must be moved farther upstream to a region outside the effect. If the boundary point is not moved, a value of flow will be forced in simulation at the external upstream boundary that could not occur at that point.
As outlined by Cunge and others (1980, p. 36), this boundary condition at the downstream end of a stream system must be applied with care. First, the imposed flow may exceed the capacity of the channel to deliver water to that node. Second, if flow is specified as a function of time at both the external upstream and downstream boundaries for a model simulation, then any errors in the values of these flows will be reflected in the water levels. In some cases, the stream system may be partially or completely dewatered because of these errors.
(122)
,
where
is the elevation of the water surface at the external
boundary node and fz(t) is the
water-surface elevation imposed at the external boundary node.
The imposed water-surface elevation may result in a depth of
flow that is too small for the flow to remain subcritical at
the external boundary. This condition is checked for in FEQ,
and critical flow is forced in the computations to prevent
supercritical flow at the external boundary. The state of the
flow at the external boundary is maintained in simulation as
for code 5, type 1 (expansion), discussed previously. If flow
at the external boundary is critical, then the control is
drowned whenever the imposed water-surface elevation exceeds
the water-surface elevation at the node at critical flow. In
the computation of a critical flow, it is assumed that
= 
=
1; however, if the cross-section table at the external boundary
node contains tabulated values of critical flow (table types 22
and 25, see section 11.1.5), then the tabulated value of
critical flow is used.