Open Channel Flow — Uniform Flow & Most Economical Sections — Fluid Mechanics & Hydraulics Notes

Last reviewed 16 Sept 2026 · 6 min read

Open channels

An open channel is a conduit in which liquid flows with a free surface at atmospheric pressure — rivers, canals, flumes, and sewers or culverts flowing partly full. The driving force is gravity (component of weight along the slope), not pressure.

Pipe flow	Open-channel flow
Flows full, under pressure	Free surface at atmospheric pressure
Driven by pressure gradient	Driven by gravity (bed slope)
HGL is above the pipe	HGL coincides with the water surface
Cross-section fixed	Flow area varies with depth

Types of channels

Prismatic — constant cross-section and bed slope (most artificial canals); non-prismatic — natural rivers.
Rigid boundary (lined) and mobile boundary (alluvial, erodible).

Types of flow

Steady / unsteady — depth at a section constant / changing with time.
Uniform / non-uniform (varied) — depth constant / changing along the channel.
Gradually varied flow (GVF) — depth changes slowly over a long distance (backwater).
Rapidly varied flow (RVF) — abrupt change over a short distance (hydraulic jump, flow over a weir).
Subcritical ( $F r < 1$ ), critical ( $F r = 1$ ), supercritical ( $F r > 1$ ), with $F r = V / g D$ .
Laminar / turbulent — based on $R e = V R / ν$ (laminar below about 500; open-channel flow in practice is almost always turbulent).

In uniform flow the depth, velocity and area are constant along the channel, and the bed, water surface and energy line are parallel ( $S_{0} = S_{w} = S_{f}$ ). The depth of uniform flow is the normal depth $y_{n}$ .

Geometric elements

Element	Definition
Flow area $A$	Cross-sectional area of flow
Wetted perimeter $P$	Length of boundary in contact with liquid
Hydraulic radius $R$	$A / P$ (hydraulic mean depth)
Top width $T$	Width of free surface
Hydraulic depth $D$	$A / T$
Section factor (critical flow)	$Z = A D$
Section factor (uniform flow)	$A R^{2/3}$

For a trapezoid (bottom width $b$ , depth $y$ , side slope $m$ H : 1V): $A = (b + m y) y$ , $P = b + 2 y 1 + m^{2}$ , $T = b + 2 m y$ .

For a wide rectangular channel ( $b ≫ y$ ): $R \approx y$ .

Velocity distribution

Velocity is zero at the bed and banks and increases towards the surface; the maximum velocity usually occurs slightly below the free surface (about 0.05–0.25 of the depth), because of air resistance and secondary currents.
Mean velocity in a vertical ≈ velocity at 0.6 depth below the surface, or the average of velocities at 0.2 and 0.8 depth (current-meter practice).
Surface velocity ≈ 1.1–1.25 × mean velocity (float measurements use a reduction factor).

Uniform flow formulas

FormulaChezy and Manning

Chezy: $V = C R S$ ( $C$ has dimensions $L^{1/2} T^{- 1}$ )

Manning: $V = \frac{1}{n} R^{2/3} S^{1/2}$ (SI units)

Relation: $C = \frac{R ^{1/6}}{n}$ ; also $C = 8 g / f$

Discharge: $Q = A V = \frac{1}{n} A R^{2/3} S^{1/2} = K S$ , where the conveyance $K = \frac{1}{n} A R^{2/3}$

Other formulas for Chezy's $C$ :

Bazin: $C = \frac{87}{1 + K / R}$ ( $K$ depends on surface roughness).
Ganguillet–Kutter: a longer expression in $n$ , $S$ and $R$ (historically used for Indian canals).

Typical Manning's $n$ (indicative): smooth cement/glass about 0.010–0.011; concrete about 0.013–0.015; brickwork about 0.015; unlined earth canals in good condition about 0.020–0.025; natural streams 0.025–0.035 and higher with weeds.

Normal depth is found by solving $A R^{2/3} = \frac{n Q}{S}$ — by trial, charts or iteration.

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Open Channel Flow — Uniform Flow & Most Economical Sections