Derivation of equation for unit-width discharge under a sluice gate

Let:

• y1 = headwater depth, i.e., flow depth upstream of sluice gate, under parallel flow.

• y2 = gate opening, or flow depth at sluice gate, under non-parallel flow.

• y3 = tailwater depth, i.e., flow depth downstream of sluice gate, under parallel flow.

The conservation of energy between headwater and tailwater depths:

• y1 + [v12/(2g)] = y3 + [v32/(2g)]

By conservation of mass:

• q = v1y1 = v3y3

• y1 + [q2/(2gy12)] = y3 + [q2/(2gy32)]

• q = y1 y3 [2g/(y1 + y3)]1/2

To express this equation in terms of the gate opening y2, a contraction coefficient is defined:

• Cc = y3/y2

• q = {Cc/[1 + Cc (y2/y1)]1/2} (2gy1)1/2 y2

Defining a discharge coefficient:

• Cd = Cc/[1 + Cc (y2/y1)]1/2

• q = Cd (2gy1)1/2 y2

Defining a reference velocity:

• v2 = Cd (2gy1)1/2

The unit-width discharge under a rectangular sluice gate is:

• q = v2 y2

The contraction coefficient Cc varies within a narrow range, between 0.598 and 0.611. A value of Cc = 0.61 has been recommended for practical use (Henderson: "Open channel flow," MacMillan, 1966).