Momentum Based Junction Method

The user can choose a momentum-based method to solve the junction problem instead of the default energy based method. As described previously, there are six possible flow conditions at the junction. The momentum-based method uses the same logic as the energy based method for solving the junction problem. The only difference is that the momentum-based method solves for the water surfaces across the junction with the momentum equation.
Also, the momentum equation is formulated such that it can take into account the angles at which reaches are coming into or leaving the junction. To use the momentum based method, the user must supply the angle for any reach who's flow lines are not parallel to the main stems flow lines. An example of a flow combining junction is shown below in the figure below. In this example, angles for both reaches 1 and 2 could be entered. Each angle is taken from a line that is perpendicular to cross-section 3.0 of reach 3.
Example Geometry for Applying the Momentum Equation to a Flow Combining Junction

For subcritical flow, the water surface is computed up to section 3.0 of reach 3 by normal standard step backwater calculations. If the momentum equation is selected, the program solves for the water surfaces at sections 4.0 and 0.0 by performing a momentum balance across the junction. The momentum balance is written to only evaluate the forces in the X direction (the direction of flow based on cross section 3.0 of reach 3). For this example the equation is as follows:

1)	$\begin{array}{l}\displaystyle SF_3 = SF_4 cos \theta _1 - F_{4-3} + W_{x_{4-3}} + SF_0 cos \theta _2 - F_{f_{0-3}} + W_{x_{0-3}}\end{array}$

Symbol	Description	Units
$\begin{array}{l}SF\end{array}$	Specific Force (as define in (Mixed Flow Regime Calculations:3))

The frictional and the weight forces are computed in two segments. For example, the friction and weight forces between sections 4.0 and 3.0 are based on the assumption that the centroid of the junction is half the distance between the two sections. The first portion of the forces are computed from section 4.0 to the centroid of the junction, utilizing the area at cross section 4.0. The second portion of the forces are computed from the centroid of the junction to section 3.0, using a flow weighted area at section 3.0. The equations to compute the friction and weight forces for this example are as follows:

Forces due to friction:

2)	$\begin{array}{l}\displaystyle F_{x_{4-3}} = \overline{S} _{f_{4-3}} \frac{L_{4-3}}{2} A_4 cos \theta _1 +\overline{S} _{f_{4-3}} \frac{L_{4-3}}{2} A_3 \frac{Q_4}{Q_3}\end{array}$

3)	$\begin{array}{l}\displaystyle F_{x_{0-3}} = \overline{S} _{f_{0-3}} \frac{L_{0-3}}{2} A_0 cos \theta _1 +\overline{S} _{f_{0-3}} \frac{L_{0-3}}{2} A_3 \frac{Q_0}{Q_3}\end{array}$

Forces due to weight of water:

4)	$\begin{array}{l}\displaystyle W_{x_{4-3}} = S_{0_{4-3}} \frac{L_{4-3}}{2} A_4 cos \theta _1 +S_{0_{4-3}} \frac{L_{4-3}}{2} A_3 \frac{Q_4}{Q_3}\end{array}$

5)	$\begin{array}{l}\displaystyle W_{x_{4-3}} = S_{f_{4-3}} \frac{L_{4-3}}{2} A_4 cos \theta _1 +S_{f_{4-3}} \frac{L_{4-3}}{2} A_3 \frac{Q_4}{Q_3}\end{array}$

To solve the momentum balance equation (Equation 4-5) for this example, the following assumptions are made:

The water surface elevations at section 4.0 and 0.0 are solved simultaneously, and are assumed to be equal to each other. This is a rough approximation, but it is necessary in order to solve (1). Because of this assumption, the cross sections around the junction should be closely spaced in order to minimize the error associated with this assumption.
The area used at section 3.0 for friction and weight forces is distributed between the upper two reaches by using a flow weighting. This is necessary in order not to double account for the flow volume and frictional area.

When evaluating supercritical flow at this type of junction (see figure above), the water surface elevations at sections 4.0 and 0.0 are computed from forewater calculations, and therefore the water surface elevations at section 3.0 can be solved directly from (1).

For mixed flow regime computations, the solution approach is the same as the energy based method, except the momentum equation is used to solve for the water surfaces across the junction.

An example of applying the momentum equation to a flow split is shown in the figure below:
Example Geometry for Applying the Momentum Equation To a Flow Split Type of Junction
For the flow split shown in in the figure above, the momentum equation is written as follows:

6)	$\begin{array}{l}\displaystyle SF_4 = SF_2 cos \theta _1 +F_{f_{4-2}} -W_{x_{4-2}} +SF_3 cos \theta _2 F_{f_{4-3}} - W_{x_{4-3}}\end{array}$

For subcritical flow, the water surface elevation is known at sections 2.0 and 3.0, and the water surface elevation at section 4.0 can be found by solving Equation 4-10. For supercritical flow, the water surface is known at section 4.0 only, and, therefore, the water surface elevations at sections 3.0 and 2.0 must be solved simultaneously. In order to solve Equation 4-10 for supercritical flow, it is assumed that the water surface elevations at sections 2.0 and 3.0 are equal.

Mixed flow regime computations for a flow split are handled in the same manner as the energy based solution, except the momentum equation (6) is used to solve for the water surface elevations across the junction.