Downstream Boundary Conditions

Downstream boundary conditions are required at the downstream end of all reaches which are not connected to other reaches or storage areas. Four types of downstream boundary conditions can be specified:

a stage hydrograph,
a flow hydrograph,
a single-valued rating curve,
Normal Depth from Manning's equation.

Stage Hydrograph. A stage hydrograph of water surface elevation versus time may be used as the downstream boundary condition if the stream flows into a backwater environment such as an estuary or bay where the water surface elevation is governed by tidal fluctuations, or where it flows into a lake or reservoir of known stage(s). At time step $\begin{array}{l}(n+1) \Delta t\end{array}$ , the boundary condition from the stage hydrograph is given by:

1)	$\begin{array}{l}\displaystyle \Delta Z_N = z_N^{n+1} - z_N^n\end{array}$

The finite difference form of (1) is:

2)	$\begin{array}{l}\displaystyle CDZ_m \Delta Z_N = CDB_m\end{array}$

Symbol	Description
$\begin{array}{l}CDZ_m\end{array}$	1
$\begin{array}{l}CDB_m\end{array}$	$\begin{array}{l}z_N^{n+1} - z_N^n\end{array}$

Flow Hydrograph. A flow hydrograph may be used as the downstream boundary condition if recorded gage data is available and the model is being calibrated to a specific flood event. At time step $\begin{array}{l}(n+1) \Delta t\end{array}$ , the boundary condition from the flow hydrograph is given by the finite difference equation:

3)	$\begin{array}{l}\displaystyle CDQ_m \Delta Q_N = CDB_m\end{array}$

Symbol	Description
$\begin{array}{l}CDQ_m\end{array}$	1
$\begin{array}{l}CDB_m\end{array}$	$\begin{array}{l}Q_N^{n+1} – Q_N^n\end{array}$

Single Valued Rating Curve. The single valued rating curve is a monotonic function of stage and flow. An example of this type of curve is the steady, uniform flow rating curve. The single valued rating curve can be used to accurately describe the stage-flow relationship of free outfalls such as waterfalls, or hydraulic control structures such as spillways, weirs or lock and dam operations. When applying this type of boundary condition to a natural stream, caution should be used. If the stream location would normally have a looped rating curve, then placing a single valued rating curve as the boundary condition can introduce errors in the solution. Too reduce errors in stage, move the boundary condition downstream from your study area, such that it no longer affects the stages in the study area. Further advice is given in (USACE, 1993).

At time $\begin{array}{l}(n+1) \Delta t\end{array}$ the boundary condition is given by:

4)	$\begin{array}{l}\displaystyle Q_N + \theta \Delta Q_N = D_{k-1} + \displaystyle \frac{D_k - D_{k-1}}{S_k - S_{k-1}} \left( z_N + \Delta z_N - S_{k-1} \right)\end{array}$

Symbol	Description
$\begin{array}{l}D_k\end{array}$	$\begin{array}{l}K^{th}\end{array}$ discharge ordinate
$\begin{array}{l}S_k\end{array}$	$\begin{array}{l}K^{th}\end{array}$ stage ordinate

After collecting unknown terms on the left side of the equation, the finite difference form of Equation 2-118 is:

5)	$\begin{array}{l}\displaystyle CDQ_m \Delta Q + CDZ_m \Delta z = CDB_m\end{array}$

Symbol	Description
$\begin{array}{l}CDQ_m\end{array}$	$\begin{array}{l}\theta\end{array}$
$\begin{array}{l}CDZ_m\end{array}$	$\begin{array}{l}\displaystyle \frac{D_k - D_{k-1}}{S_k - S_{k-1}}\end{array}$
$\begin{array}{l}CDB_m\end{array}$	$\begin{array}{l}Q_N + D_{k-1} + \displaystyle \frac{D_k - D_{k-1}}{S_k - S_{k-1}} \left( z_N - S_{k-1} \right)\end{array}$

Normal Depth. Use of Manning's equation with a user entered friction slope produces a stage considered to be normal depth if uniform flow conditions existed. Because uniform flow conditions do not normally exist in natural streams, this boundary condition should be used far enough downstream from your study area that it does not affect the results in the study area. Manning's equation may be written as:

6)	$\begin{array}{l}\displaystyle Q = K \left( S_f \right) ^{0.5}\end{array}$

where: $\begin{array}{l}K\end{array}$ represents the conveyance and $\begin{array}{l}S_f\end{array}$ is the friction slope.