Hydraulic Jump and Flow Measurement

When fast, shallow (supercritical) flow meets slower, deeper flow, it changes abruptly in a hydraulic jump, losing energy. Jumps are used below spillways to protect the river bed.

Hydraulic jump

Using momentum conservation in a rectangular channel, the sequent (conjugate) depths y₁ (before) and y₂ (after) satisfy:

y₂/y₁ = ½ (−1 + √(1 + 8Fr₁²))

• Energy loss: ΔE = (y₂ − y₁)³/(4 y₁ y₂).
• Length of jump ≈ 6(y₂ − y₁) on a horizontal floor.
• Height of jump = y₂ − y₁.

Weirs and notches

• Rectangular weir (crest length L, head H): Q = (2/3) C_d √(2g) L H^(3/2). With end contractions (Francis): L_eff = L − 0.1nH.
• Triangular (V) notch with angle θ: Q = (8/15) C_d √(2g) tan(θ/2) H^(5/2). Good for small flows, since Q changes fast with H.
• Cipolletti (trapezoidal, side slopes 1 horizontal : 4 vertical): compensates for end contractions.
• Broad-crested weir: critical depth forms on the crest; Q = 1.705 C_d L H^(3/2) (SI units).
• Error: a small error dH in head gives dQ/Q = (3/2)(dH/H) for a rectangular weir and (5/2)(dH/H) for a V-notch.

Other devices

• Venturi flume / Parshall flume: a constriction forcing critical flow; good for canals with sediment.
• Orifice: Q = C_d a √(2gH), with C_d = C_c C_v (about 0.62).
• Sluice gate: free flow below a gate creates supercritical flow, often followed by a jump.

Worked example

Flow at y₁ = 0.5 m and V₁ = 6 m/s enters a jump (g = 9.81).

• Fr₁ = 6/√(9.81 × 0.5) = 2.71.
• y₂ = 0.25 × (−1 + √(1 + 8 × 7.34)) = 0.25 × (−1 + 7.73) = 1.68 m.
• ΔE = (1.18)³/(4 × 0.5 × 1.68) = 0.49 m.

Practice