Added Force Term

The friction and pressure forces from the banks do not always describe all the forces that act on the water. Structures such as bridge piers, navigation dams, and cofferdams constrict the flow and exert additional forces, which oppose the flow. In localized areas these forces can predominate and produce a significant increase in water surface elevation (called a "swell head") upstream of the structure.

For a differential distance, $\begin{array}{l}dx\end{array}$ , the additional forces in the contraction produce a swell head of $\begin{array}{l}dh_l\end{array}$ . This swell head is only related to the additional forces. The rate of energy loss can be expressed as a local slope:

1)	$\begin{array}{l}\displaystyle \displaystyle S_h=\frac{dh_l}{dx}\end{array}$

The friction slope in (.Momentum Equation0 v6.1:11) can be augmented by this term:

2)	$\begin{array}{l}\displaystyle \displaystyle \frac{\partial Q}{\partial t}+ \frac{\partial (VQ)}{\partial x} +gA \left( \frac{\partial z_s}{\partial x} + S_f + S_h \right) =0\end{array}$

For steady flow, there are a number of relationships for computation of the swell head upstream of a contraction. For navigation dams, the formulas of Kindsvater and Carter, d'Aubuisson (Chow, 1959), and Nagler were reviewed by Denzel (1961). For bridges, the formulas of Yarnell (WES, 1973) and the Federal Highway Administration (FHWA, 1978) can be used. These formulas were all determined by experimentation and can be expressed in the more general form:

3)	$\begin{array}{l}\displaystyle \displaystyle h_l=C \frac{V^2}{2g}\end{array}$

where $\begin{array}{l}h_l\end{array}$ is the head loss and $\begin{array}{l}C\end{array}$ is a coefficient. The coefficient $\begin{array}{l}C\end{array}$ is a function of velocity, depth, and the geometric properties of the opening, but for simplicity, it is assumed to be a constant. The location where the velocity head is evaluated varies from method to method. Generally, the velocity head is evaluated at the tailwater for tranquil flow and at the headwater for supercritical flow in the contraction.

If $\begin{array}{l}h_l\end{array}$ occurs over a distance $\begin{array}{l}\Delta x_e\end{array}$ , then $\begin{array}{l}h_l = \overline{S}_h \Delta x_e\end{array}$ and $\begin{array}{l}\overline{S}_h = h_l / \Delta x_e\end{array}$ where $\begin{array}{l}\overline{S}_h\end{array}$ is the average slope over the interval $\begin{array}{l}\Delta x_e\end{array}$ . Within HEC-RAS, the steady flow bridge and culvert routines are used to compute a family of rating curves for the structure. During the simulation, for a given flow and tailwater, a resulting headwater elevation is interpolated from the curves. The difference between the headwater and tailwater is set to $\begin{array}{l}h_l\end{array}$ and then $\begin{array}{l}\overline{S}_h\end{array}$ is computed. The result is inserted in the finite difference form of the momentum equation (Equation 2-93), yielding:

4)	$\begin{array}{l}\displaystyle \displaystyle \frac{\Delta (Q_c \Delta x_c + Q_f \Delta x_f)}{\Delta t \Delta x_e} + \frac{\Delta (\beta VQ)}{\Delta x_e} + g \overline{A} \left( \frac{\Delta z_s}{\Delta x_e} + \overline{S}_f + \overline{S}_h \right) = 0\end{array}$