Specific Energy & Critical Depth

OpenHydroLab · Rectangular flume, b = 0.20 m

Experiment

Sluice gate — set y₁ & a (lab setup) Sluice gate — hold Q, vary a (lab step 4) Flow over a hump — choking

Controls

Upstream depth y₁

Gate opening a

Show

Water particles (u = q/h, continuity) Energy grade line Critical-depth line y꜀ Second discharge curve (y₁ = 150 mm run)

Aha demo: switch to Hold Q and press Sweep — the E–y curve stays fixed (same q) while y₁ and y₂ slide along it and meet at the critical point as a → y꜀.

Flow

Specific discharge q
Discharge Q = q·b
V₁ = q/y₁
Fr₁ = V₁/√(gy₁)
V₂ = q/y₂
Fr₂ = V₂/√(gy₂)

y₁ SUB

y₂ SUPER

Energy & critical depth

E₁ = y₁ + q²/(2gy₁²)
E₂ = y₂ + q²/(2gy₂²)
Critical depth y꜀ = (q²/g)^⅓
E_min = 1.5·y꜀

E = y + q²/(2gy²) · y꜀ = (q²/g)^⅓
q = y₂·√( 2gy₁ / (1 + y₂/y₁) ) (E₁ = E₂, y₂ ≈ a)

Sluice gate rating (lab Eq. 10–11)

Q_ideal = b·a·√(2gy₁)
C_d = Q/Q_ideal
Energy check E₁ − E₂

Frictionless gate: specific energy is conserved across the gate, so y₁ (subcritical) and y₂ (supercritical) are alternate depths — the same vertical line on the E–y diagram. y₂ ≈ a as in the lab manual.

Hump height Δz
Choking limit Δz_crit = E₁ − E_min
Depth at crest
Surface drop at crest
Upstream rise (choked)
Energy budget max\|E+z−E₁\|