Equations for Basic Profile Calculations

Water surface profiles are computed from one cross section to the next by solving the Energy equation with an iterative procedure called the standard step method. The Energy equation is written as follows:

1)	$\begin{array}{l}\displaystyle Z_2+Y_2+\frac{\alpha_2 V_2^2}{2g} = Z_1+Y_1+\frac{\alpha_1 V_1^2}{2g}+h_e\end{array}$

Symbol	Description	Units
$\begin{array}{l}Z_1, Z_2\end{array}$	elevation of the main channel inverts
$\begin{array}{l}Y_1, Y_2\end{array}$	depth of water at cross sections
$\begin{array}{l}V_1, V_2\end{array}$	average velocities (total discharge/ total flow area)
$\begin{array}{l}a_1, a_2\end{array}$	velocity weighting coefficients
$\begin{array}{l}g\end{array}$	gravitational acceleration
$\begin{array}{l}h_e\end{array}$	energy head loss

A diagram showing the terms of the energy equation is shown in the figure below.

The energy head loss $\begin{array}{l}(h_e)\end{array}$ between two cross sections is comprised of friction losses and contraction or expansion losses. The equation for the energy head loss is as follows:

2)	$\begin{array}{l}\displaystyle h_e=L\overline{S}_f+C \displaystyle\left( \frac{\alpha_2 V_2^2}{2g}-\frac{\alpha_1 V_1^2}{2g} \right)\end{array}$

Symbol	Description	Units
$\begin{array}{l}L\end{array}$	discharge weighted reach length
$\begin{array}{l}\overline{S}_f\end{array}$	representative friction slope between two sections
$\begin{array}{l}C\end{array}$	expansion or contraction loss coefficient

The discharge weighted reach length, $\begin{array}{l}L\end{array}$ , is calculated as:

3)	$\begin{array}{l}\displaystyle L=\frac{L_{lob}\ \overline{Q}_{lob}+L_{ch}\ \overline{Q}_{ch}+L_{rob}\ \overline{Q}_{rob}}{\overline{Q}_{lob}+\overline{Q}_{ch}+\overline{Q}_{rob}}\end{array}$

Symbol	Description	Units
$\begin{array}{l}L_{lob}, L_{ch}, L_{rob}\end{array}$	cross section reach lengths specified for flow in the left overbank, main channel, and right overbank, respectively
$\begin{array}{l}\overline{Q_{lob}} + \overline{Q_{ch}} + \overline{Q_{rob}}\end{array}$	arithmetic average of the flows between sections for the left overbank, main channel, and right overbank, respectively