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).
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