Normal Depth Calculator

Depth of uniform flow in a prismatic channel — solved by bisection on Manning's equation Q = (1/n) × A × R_h^2/3 × S^1/2. Supports rectangular, trapezoidal, and circular channels with mild / steep / critical slope classification.

Open Channel Flow · Hydraulics

Inputs

Solve for:

Channel cross-section:

Bottom width b (m)

Flow rate Q (m³/s)

Manning's roughness n (dimensionless)

Channel bed slope S (m/m)

Theory

Normal depth definition:

Q = (1/n) × A(y_n) × R_h(y_n)^2/3 × S^1/2

y_n is found by bisection — no closed form except for wide channels

Normal depth is the depth at which steady uniform flow occurs — the gravitational component along the slope exactly balances the friction force. It depends on Q, n, S, and channel geometry, but NOT on conditions upstream or downstream.

Slope classification:

Slope type	Condition	Normal flow regime
Mild	S < S_c, y_n > y_c	Subcritical (Fr < 1)
Critical	S = S_c, y_n = y_c	Critical (Fr = 1)
Steep	S > S_c, y_n < y_c	Supercritical (Fr > 1)
Horizontal	S = 0	No normal depth exists
Adverse	S < 0 (uphill)	No normal depth exists

Channel geometry formulas:

Rectangular (b = bottom width):

A = b × y P = b + 2y R = by/(b + 2y)

Trapezoidal (z = H:V side slope):

A = (b + zy)y P = b + 2y√(1+z²) R = A/P

Circular (D = diameter, θ = 2·arccos(1 − 2y/D)):

A = (D²/8)(θ − sin θ) P = Dθ/2 R = A/P