Relationship between critical depth and discharge coefficient for free overflow on dam spillway

In general, the discharge for free overflow on dam spillway can be expressed as follow:

$\begin{equation} Q=C\cdot B\cdot H^{3/2} \end{equation}$

$\begin{align} &Q & & \text{discharge}\\ &C & & \text{discharge coefficient}\\ &B & & \text{length of overflow crest}\\ &H & & \text{design head including approach velocity head} \end{align}$

On the other hand, a critical depth of rectangular cross section channel can be expressed as follow:

$\begin{equation} h_c=\left(\cfrac{Q^2}{g\cdot B^2}\right)^{1/3} \end{equation}$

$\begin{align} &h_c & & \text{critical depth}\\ &Q & & \text{discharge}\\ &B & & \text{length of overflow crest}\\ &g & & \text{gravity acceleration} \end{align}$

When it is assumed the water depth at the overflow crest of the dam is equal to the critical depth, the design head including approach velocity head can be expressed as follow:

$\begin{equation} H=h_c+\cfrac{1}{2g}\left(\cfrac{Q}{B\cdot h_c}\right)^2 \end{equation}$

Considering a unit length of overflow crest which means $B=1.0 m$ ,

$\begin{equation} Q=C\cdot H^{3/2} \qquad H=h_c+\cfrac{1}{2g}\left(\cfrac{Q}{h_c}\right)^2 \qquad h_c=\left(\cfrac{Q^2}{g}\right)^{1/3} \end{equation}$