Specific energy, critical depth and the Froude number
Why one discharge can flow at two depths, what critical flow is, and how the Froude number separates subcritical from supercritical flow.
Specific energy is the energy per unit weight measured from the channel bed:
E = y + V²2 g = y + q²2 g y²
where y is depth and q = Q/b is the discharge per metre width of a rectangular channel.
Two depths, one energy
For a given q, plotting E against y gives a curve with a minimum. Above the minimum, each value of E can be reached at two depths, called alternate depths: a deep, slow (subcritical) one and a shallow, fast (supercritical) one.
Critical depth
At the minimum point the flow is critical. For a rectangular channel:
yc = ( q²g )1/3 Emin = 32 yc Vc = √(g yc)
Froude number
Fr = V√(g D)
D is the hydraulic depth, A/T (equal to y for a rectangle).
| Fr | Flow type | What it looks like |
|---|---|---|
| below 1 | Subcritical | Deep and slow; disturbances travel upstream |
| = 1 | Critical | Unstable, wavy surface |
| above 1 | Supercritical | Shallow and fast; nothing travels upstream |
Worked example
A 4 m wide rectangular channel carries 12 m³/s. So q = 3 m³/s per m.
yc = ( 3²9.81 )1/3 = 0.97 m Emin = 1.46 m
If the actual depth is 1.8 m, V = 3/1.8 = 1.67 m/s and Fr = 1.67/√(9.81 × 1.8) = 0.40, so the flow is subcritical.
Why it matters
Critical depth occurs at a free overfall, over a broad-crested weir and at a sudden steepening of slope. Engineers use these control points to measure flow and to start water surface profile calculations.
Quick check
In a rectangular channel at critical depth, the specific energy is: