Open Channel Flow: Uniform Flow

Open channel flow implies flow in a conduit where the liquid surface is exposed to atmospheric pressure (free surface). Examples include rivers, canals, sewers, and flumes.

Velocity is not uniform across a channel. It is zero at the solid boundaries due to friction (no-slip condition) and maximum near the free surface.

• Maximum Velocity: Usually occurs slightly below the free surface (about $0.05y$ to $0.25y$ depth) due to secondary currents and surface tension, rather than exactly at the surface.
• Mean Velocity ($V$): For practical engineering, a single average velocity is used, calculated as $V = Q/A$.
• Measurement: In field hydrology, the mean velocity of a vertical section is often approximated by taking the average of the velocities measured at $0.2$ and $0.8$ of the depth, or just the velocity at $0.6$ of the depth from the surface.

To analyze open channel flow, we define key geometric properties of the cross-section.

• Depth of Flow ($y$): Vertical distance from the channel bottom to the free surface.
• Top Width ($T$): Width of the channel at the free surface.
• Wetted Perimeter ($P$): The length of the channel boundary in contact with the fluid.
• Hydraulic Radius ($R$): The ratio of the flow area to the wetted perimeter.
• Hydraulic Depth ($D$): The ratio of the flow area to the top width ($D = A/T$).

Uniform flow occurs when the flow depth, area, and velocity remain constant along the channel. This implies that the energy line slope ($S_f$), water surface slope ($S_w$), and channel bottom slope ($S_0$) are all equal ($S_f = S_w = S_0$).

The Chezy formula is one of the earliest empirical formulas developed to describe uniform flow velocity in an open channel.

The Chezy coefficient ($C$) is not a constant; it depends on the roughness of the channel and the hydraulic radius. Several empirical formulas exist to determine $C$, including Manning's relation for $C$, Bazin's formula, and Kutter's formula.

Kutter's formula is a more complex empirical equation historically used before Manning became standard. It expresses the Chezy coefficient $C$ as a function of the slope ($S$), hydraulic radius ($R$), and the roughness factor ($n$).

The Manning formula is the most widely used empirical formula for open channel flow analysis. It relates the uniform velocity to the hydraulic radius, channel slope, and boundary roughness.

• Manning's Roughness Coefficient ($n$):
  • Clean concrete: $n \approx 0.013$
  • Earth channel: $n \approx 0.022$
  • Natural stream: $n \approx 0.035 - 0.050$

Interact with the simulation below to see how geometric parameters and roughness affect uniform flow in an open channel.

The "most efficient" or "best hydraulic" cross-section is the one that conveys the maximum discharge for a given area, slope, and roughness. This corresponds to the section with the minimum wetted perimeter ($P$).

To achieve the most efficient hydraulic section, the cross-sectional geometry is optimized to minimize the wetted perimeter $P$ for a given flow area $A$. This minimizes friction losses and maximizes velocity and discharge.