Specific Energy & Critical Flow

Specific energy and the specific energy diagram, alternate depths; critical depth, critical velocity and minimum specific energy for rectangular, triangular, parabolic and general sections; Froude number and subcritical/supercritical flow; discharge diagram and maximum discharge; specific force (momentum function); critical slope, mild and steep slopes; transitions — humps and width contractions, choking; critical-depth flumes (Venturi and Parshall) — with solved numericals.

Specific Energy & Critical Flow

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📑 Contents (9 sections)

Last reviewed 16 Sept 2026 · 6 min read

Specific energy

Specific energy $E$ is the energy per unit weight of liquid measured with respect to the channel bed:

E = y + \frac{V ^{2}}{2 g} = y + \frac{Q ^{2}}{2 g A ^{2}}

For a rectangular channel with discharge per unit width $q$ :

E = y + \frac{q ^{2}}{2 g y ^{2}}

Unlike total energy, specific energy can increase or decrease along a channel (bed level is the datum at each section).

Specific energy diagram

Plotting $y$ against $E$ for constant $q$ :

The curve approaches the line $E = y$ (45°) for large depths and the $E$ -axis for small depths.
For any $E$ greater than the minimum, there are two possible depths — alternate depths — one subcritical ( $y > y_{c}$ ) and one supercritical ( $y < y_{c}$ ).
At the minimum specific energy, the two depths coincide — the critical depth $y_{c}$ .

Critical flow

Critical flow is the state of minimum specific energy for a given discharge (equivalently, maximum discharge for a given specific energy).

FormulaCritical flow — general section

\frac{Q ^{2} T}{g A ^{3}} = 1 ⟺ \frac{V ^{2}}{g D} = 1, F r = \frac{V}{g D} = 1

Section factor: $Z = A A / T = \frac{Q}{g}$

Velocity head at critical flow = half the hydraulic depth: $\frac{V _{c}^{2}}{2 g} = \frac{D _{c}}{2}$

FormulaCritical depth for common sections

Section	Critical depth	$E_{min}$
Rectangular	$y_{c} = (\frac{q ^{2}}{g})^{1/3}$ ; $V_{c} = g y_{c}$	$\frac{3}{2} y_{c}$
Triangular (side slope $m$ )	$y_{c} = (\frac{2 Q ^{2}}{g m ^{2}})^{1/5}$	$\frac{5}{4} y_{c}$
Parabolic	—	$\frac{4}{3} y_{c}$

Subcritical, critical and supercritical flow

	Subcritical (tranquil)	Critical	Supercritical (rapid, shooting)
Froude number	$F r < 1$	$F r = 1$	$F r > 1$
Depth	$y > y_{c}$	$y = y_{c}$	$y < y_{c}$
Velocity	$V < g D$	$V = g D$	$V > g D$
Small surface waves	Travel upstream and downstream	Standing wave	Travel only downstream
Control	Downstream	—	Upstream

$g D$ is the celerity of a small gravity wave in shallow water.

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