Normal and critical flow

Where its roughness changes, the section is divided into parts, and each carries its own share of the discharge.

Normal depth by Manning

At any water level, Manning’s equation in SI units gives each part’s discharge, and the section carries their sum:

$$Q = \sum_j \frac{1}{n_j} A_j R_j^{2/3} S^{1/2}, \qquad R_j = \frac{A_j}{P_j}$$

The vertical lines between parts are not wetted perimeter. Divided this way, each part’s conveyance grows as the water rises, so the discharge does too, and every discharge has one normal depth. A floodplain with its channel’s $n$ is not divided from it, and there the discharge can drop as the water first spreads over the floodplain: more than one depth then carries the same $Q$, and Rischio warns of it (see the rating curve).

The normal depth is the depth at which this discharge equals $Q$. Brent’s method finds it: it brackets the root and needs no derivative, so kinks in the section don’t stall it. The bracket starts from $10^{-9}$ m to 1 m above the bed, and the depth of its upper end doubles until the discharge there reaches $Q$. The tolerance is about $10^{-12}$ m.

Critical depth and the Froude number

The Froude number, $Fr = V / \sqrt{g D}$, uses the hydraulic depth $D = A/T$, not the flow depth. They are equal in a rectangle; in other shapes, $D$ is smaller. $Fr < 1$ is subcritical flow, $Fr > 1$ supercritical and $Fr = 1$ critical. $0.8 < Fr < 1.2$ raises a near-critical warning.

The critical depth is the depth at which $Q$ would flow with $Fr = 1$, where $Q^2 T / (g A^3) = 1$. Brent’s method solves it too: the bracket starts from $10^{-6}$ m to 1 m above the bed, and the depth of its upper end doubles while $Q^2 T / (g A^3) > 1$. It is reported only if its water level stays within the lower bank.

Specific energy

The specific energy is measured from the bed: its $y$ is the normal depth, not an elevation, so $E$ is not the total head. The velocity head uses the mean velocity ($\alpha = 1$).