Uniform & Non-Uniform Flow

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.

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 ($Fr<1$), critical ($Fr=1$), supercritical ($Fr>1$), with $Fr=V/\sqrt{gD}$.
• Laminar / turbulent — based on $Re=VR/\nu$ (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

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

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

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

Other formulas for Chezy's $C$:

• Bazin: $C=\frac{87}{1+K/\sqrt{R}}$ ($K$ 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 $A{R}^{2/3}=\frac{nQ}{\sqrt{S}}$ — by trial, charts or iteration.

Gradually varied flow

Gradually varied flow (GVF) is steady non-uniform flow in which the depth changes gradually over a long distance — for example the backwater curve upstream of a dam or weir.

Assumptions

1. Flow is steady; streamlines are practically parallel, so pressure distribution is hydrostatic.
2. The head loss at a section equals that of uniform flow with the same depth and velocity — Manning or Chezy formulas apply using the local depth.
3. Channel slope is small; the channel is prismatic.
4. Velocity distribution is fixed (α constant); roughness is constant along the reach.

Dynamic equation of GVF

• $dy/dx>0$: depth increases in the flow direction (backwater / rising curve).
• $dy/dx<0$: depth decreases (drawdown / falling curve).
• $dy/dx$ = 0: uniform flow ($y=y_n$).
• $dy/dx\to \infty$ as $y\to y_c$: the surface meets the critical depth line vertically in theory — in reality a hydraulic jump (rising) or a free fall/rapid drawdown occurs; GVF assumptions break down there.

Classification of channel slopes