Lateral Influx of Momentum

At stream junctions, the momentum as well as the mass of the flow from a tributary enters the receiving stream. If this added momentum is not included in the momentum equation, the entering flow has no momentum and must be accelerated by the flow in the river. The lack of entering momentum causes the convective acceleration term, $\begin{array}{l}\delta (VQ) / \delta x\end{array}$ , to become large. To balance the spatial change in momentum, the water surface slope must be large enough to provide the force to accelerate the fluid. Thus, the water surface has a drop across the reach where the flow enters creating backwater upstream of the junction on the main stem. When the tributary flow is large in relation to that of the receiving stream, the momentum exchange may be significant. The confluence of the Mississippi and Missouri Rivers is such a juncture. During a large flood, the computed decrease in water surface elevation over the Mississippi reach is over 0.5 feet if the influx of momentum is not properly considered.

The entering momentum is given by:

1)	$\begin{array}{l}\displaystyle \displaystyle M_l = \xi \frac{Q_l V_l}{\Delta x}\end{array}$

Symbol	Description	Units
$\begin{array}{l}Q_l\end{array}$	lateral inflow
$\begin{array}{l}V_l\end{array}$	average velocity of lateral inflow
$\begin{array}{l}\xi\end{array}$	fraction of the momentum entering the receiving stream

The entering momentum is added to the right side of (.Added Force Term v6.1:4), hence:

2)	$\begin{array}{l}\displaystyle \displaystyle \frac{\Delta (Q_c \Delta x_c + Q_f \Delta x_f)}{\Delta t \Delta x_e} + \frac{\Delta (\beta VQ)}{\Delta x_e} + g \overline{A} \left( \frac{\Delta z_s}{\Delta x_e} + \overline{S}_f + \overline{S}_h \right) = \xi \frac{Q_l V_l}{\Delta x}\end{array}$

(2) is only used at stream junctions in a dendritic model.