Flow through partially and fully submerged orifice

A submerged orifice discharges below the surface of the liquid on the downstream side. It may be fully or partially submerged.

Fully submerged orifice

Let the upstream and downstream liquid surfaces be at heights $H_1$ and $H_2$ above the centre of the orifice, and $H = H_1 - H_2$. Apply Bernoulli between a point in the upstream tank (negligible velocity) and the vena contracta, where the pressure is the downstream hydrostatic pressure $\gamma H_2$:

$$H_1 + 0 = H_2 + \frac{V_c^2}{2g}\;\Rightarrow\; V_c = \sqrt{2g\,(H_1 - H_2)} = \sqrt{2gH}$$

With orifice area $a$ and coefficient of contraction $C_c$, $Q = C_c a V_c$. Including the velocity coefficient $C_v$ gives $C_d = C_cC_v$:

$$Q = C_d\,a\sqrt{2g\,(H_1 - H_2)}$$

The discharge depends only on the head difference.

Partially submerged orifice

Here the downstream level lies between the top and bottom edges of the orifice, and the flow is the sum of two parts.

Let $b$ be the width of the (rectangular) orifice, $H_1$ and $H_2$ the depths from the upstream surface to the top and bottom edges, and $H$ the difference between the upstream and downstream levels ($H_1 < H < H_2$).

• Free part (top edge to the downstream level): the pressure outside is atmospheric, so it behaves like a free rectangular orifice with the head varying from $H_1$ to $H$:

$$Q_1 = \frac{2}{3}C_d b\sqrt{2g}\left[H^{3/2} - H_1^{3/2}\right]$$

• Submerged part (downstream level to bottom edge): height $(H_2 - H)$ under the constant head difference $H$:

$$Q_2 = C_d\,b\,(H_2 - H)\sqrt{2gH}$$

Total discharge:

$$Q = \frac{2}{3}C_d b\sqrt{2g}\left[H^{3/2} - H_1^{3/2}\right] + C_d b\,(H_2 - H)\sqrt{2gH}$$

 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ upstream level
   |    |    |
   |    |    H1
   |    |    v      ____  top edge
   |    H          |    |  free part
   H2   |          |    |
   |    v ~~~~~~~~~|~~~~|~~~~~~~ downstream level
   |               |    |  submerged part
   v               |____|  bottom edge