Open Channel Flow: Uniform Flow

Flow in open channels, geometric elements, Chezy and Manning formulas, and most efficient hydraulic sections.

Concept Overview

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 Distribution

How fluid velocity varies within the channel cross-section.

Velocity Profiles in Channels

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 (VV): For practical engineering, a single average velocity is used, calculated as V=Q/AV = 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.

Geometric Elements

Concept Overview

To analyze open channel flow, we define key geometric properties of the cross-section.
  • Depth of Flow (yy): Vertical distance from the channel bottom to the free surface.
  • Top Width (TT): Width of the channel at the free surface.
  • Wetted Perimeter (PP): The length of the channel boundary in contact with the fluid.
  • Hydraulic Radius (RR): The ratio of the flow area to the wetted perimeter.

    Concept Overview

    How fluid velocity varies within the channel cross-section.

    R=APR = \frac{A}{P}
  • Hydraulic Depth (DD): The ratio of the flow area to the top width (D=A/TD = A/T).

Uniform Flow Formulas

Uniform flow occurs when the flow depth, area, and velocity remain constant along the channel. This implies that the energy line slope (SfS_f), water surface slope (SwS_w), and channel bottom slope (S0S_0) are all equal (Sf=Sw=S0S_f = S_w = S_0).

Chezy Formula

One of the earliest formulas for uniform flow.

Chezy Formula

V=CRSV = C \sqrt{RS}

Variables

SymbolDescriptionUnit
VVMean velocity-
CCChezy coefficientrelated to roughness
RRHydraulic radius-
SSSlope of the energy line-

Chezy Coefficient (CC) Dependencies

Estimating the Chezy Coefficient

The Chezy coefficient (CC) is not a constant; it depends on the roughness of the channel and the hydraulic radius. Several empirical formulas exist to determine CC.
  • Manning's Equation for CC: By equating the Manning and Chezy velocity formulas (CRS=1nR2/3SC\sqrt{RS} = \frac{1}{n}R^{2/3}\sqrt{S}), we get the most common expression for CC in SI units:

    Estimating the Chezy Coefficient

    C=1nR1/6C = \frac{1}{n} R^{1/6}
  • Bazin's Formula: An older empirical formula where CC depends on an experimental roughness parameter mm:
    C=871+mRC = \frac{87}{1 + \frac{m}{\sqrt{R}}}
  • Kutter's Formula: A more complex empirical formula historically used before Manning became standard, depending on slope (SS), hydraulic radius (RR), and a roughness factor nn.

Manning Formula

The most widely used empirical formula for open channel flow.

Manning Formula

V=1nR2/3S1/2(SI Units)V = \frac{1}{n} R^{2/3} S^{1/2} \quad (\text{SI Units})
V=1.486nR2/3S1/2(English Units)V = \frac{1.486}{n} R^{2/3} S^{1/2} \quad (\text{English Units})
  • nn = Manning's roughness coefficient.
    • Clean concrete: n≈0.013n \approx 0.013
    • Earth channel: n≈0.022n \approx 0.022
    • Natural stream: n≈0.035−0.050n \approx 0.035 - 0.050
Discharge (QQ):
Q=AV=1nAR2/3S1/2Q = AV = \frac{1}{n} A R^{2/3} S^{1/2}

Open Channel Flow (Manning's Equation)

Normal Depth (y_n) = 0.000 m

Q=1nAR2/3S1/2Q = \frac{1}{n} A R^{2/3} S^{1/2}
Adjust the parameters to see how they affect the normal depth required to convey the given discharge.

Most Efficient Hydraulic Section

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 (PP).

Conditions for Efficiency

Procedure

  1. Rectangular Channel:
    • Width = 2 ×\times Depth (b=2yb = 2y).
    • Hydraulic Radius R=y/2R = y/2.
  2. Trapezoidal Channel:
    • Half-hexagon shape.
    • Top width = Sum of side lengths (T=2×side lengthT = 2 \times \text{side length}).
    • Hydraulic Radius R=y/2R = y/2.
    • Side slope angle θ=60∘\theta = 60^\circ is optimal.
  3. Circular Channel:
    • Semicircle (y=D/2y = D/2) is most efficient for open channel flow.
Key Takeaways
  • Velocity Distribution: Is non-uniform, with the maximum velocity occurring slightly below the free water surface, not on it.
  • The Hydraulic Radius (RR) is a critical parameter in open channel flow, defined as the ratio of the cross-sectional area to the wetted perimeter.
  • Unlike pipes flowing full, where R=D/4R = D/4, the hydraulic radius in open channels varies with depth.
  • Uniform flow occurs when the depth and velocity are constant along the channel, meaning the gravitational forces driving the flow exactly balance the frictional resistance from the channel boundaries.
  • The Manning Formula is the universally accepted empirical equation for uniform open channel flow.
  • The roughness coefficient (nn) significantly impacts flow; a rougher channel requires a steeper slope or larger area to convey the same discharge.
  • The most efficient hydraulic section is the geometry that provides the maximum discharge for a given cross-sectional area.
  • This is achieved by minimizing the wetted perimeter, which minimizes frictional resistance.
  • For a rectangle, the optimal shape is half a square (width = 2 ×\times depth). For a trapezoid, it's half a regular hexagon.
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