11.2 Open-Channel Flow and Manning's Equation

11.2 Open-Channel Flow and Manning's Equation

Fundamentals of Open-Channel Flow

Open-channel flow occurs when liquid flows in a conduit with a free surface exposed to atmospheric pressure. Unlike pressure flow in pipes, open-channel flow is driven solely by gravity and the slope of the channel bed. In highway design, open channels include roadside ditches, gutters, median swales, culverts flowing under gravity, and natural streams.

To analyze open-channel hydraulics, we define several geometric parameters based on the cross-sectional shape of the water flow:

• Flow Depth ($y$): The vertical distance from the lowest point of the channel bed to the water surface.
• Cross-Sectional Area ($A$): The area of flow perpendicular to the direction of velocity.
• Wetted Perimeter ($P$): The length of the channel boundary in contact with the water.
• Hydraulic Radius ($R$): The ratio of flow area to wetted perimeter ($R = A/P$). It represents the relative efficiency of the channel cross-section.
• Top Width ($T$): The width of the channel at the free water surface.
• Hydraulic Depth ($D$): The ratio of flow area to top width ($D = A/T$).

Summary of Geometric Equations

The table below summarizes the geometric relations for common channel shapes:

(where $b$ is bottom width, $z$ is side slope horizontal-to-vertical ratio $z$:1, and $d$ is diameter).

Manning's Equation for Uniform Flow

Under uniform flow conditions, the water depth, cross-sectional area, velocity, and discharge remain constant along the channel. In this state, the gravitational driving force is exactly balanced by the frictional resistance of the channel boundaries. Uniform flow is analyzed using Manning's Equation:

US Customary Units

$Q = \frac{1.486}{n} \cdot A \cdot R^{2/3} \cdot S^{1/2}$ $V = \frac{1.486}{n} \cdot R^{2/3} \cdot S^{1/2}$

SI Units

$Q = \frac{1.0}{n} \cdot A \cdot R^{2/3} \cdot S^{1/2}$ $V = \frac{1.0}{n} \cdot R^{2/3} \cdot S^{1/2}$

Where:

• $Q$ = Flow rate or discharge (cfs or $m^3/s$)
• $V$ = Mean flow velocity (fps or m/s)
• $n$ = Manning's roughness coefficient (dimensionless, representing boundary friction)
• $A$ = Cross-sectional area of flow ($ft^2$ or $m^2$)
• $R$ = Hydraulic radius (ft or m)
• $S$ = Channel slope (ft/ft or m/m)

Manning's Roughness Coefficient ($n$)

Manning's $n$ varies significantly based on the channel material and vegetation:

Specific Energy and Critical Flow

Specific energy ($E$) is the energy of flow at a given cross-section relative to the channel bed: $E = y + \frac{V^2}{2g} = y + \frac{Q^2}{2g A^2}$ For a constant discharge $Q$, the specific energy curve exhibits a minimum value at a unique depth called the critical depth ($y_c$).

The Froude Number

The Froude number ($Fr$) is a dimensionless parameter representing the ratio of inertial forces to gravitational forces: $Fr = \frac{V}{\sqrt{g D}}$ where $g$ is the acceleration of gravity ($32.2 \text{ ft/s}^2$ or $9.81 \text{ m/s}^2$) and $D$ is the hydraulic depth ($A/T$).

The Froude number determines the flow regime:

1. Subcritical Flow ($Fr < 1$): Water depth is greater than critical depth ($y > y_c$). Flow is slow, tranquil, and controlled by downstream conditions.
2. Critical Flow ($Fr = 1$): Flow is at minimum specific energy ($y = y_c$).
3. Supercritical Flow ($Fr > 1$): Water depth is less than critical depth ($y < y_c$). Flow is fast, rapid, and controlled by upstream conditions. A transition from supercritical to subcritical flow results in a hydraulic jump, which dissipates energy but can cause severe erosion.

Critical Depth in Rectangular Channels

For a rectangular channel, the critical depth ($y_c$) is solved directly as: $y_c = \left(\frac{q^2}{g}\right)^{1/3}$ where $q = Q/b$ is the unit discharge (discharge per unit width of the channel, cfs/ft).

Channel Lining and Shear Stress Design (HEC-15)

Roadside ditches must be designed to withstand erosion caused by flowing water. The Federal Highway Administration (FHWA) publication HEC-15 governs the design of flexible channel linings (e.g., grass, gravel, riprap) using tractive force theory.

The average shear stress ($\tau$) exerted by flowing water on the channel boundary is: $\tau = \gamma \cdot R \cdot S$ For wide channels ($b/y > 10$), the hydraulic radius $R$ approaches the flow depth $y$, simplifying the maximum boundary shear stress on the channel bed to: $\tau_d = \gamma \cdot y \cdot S$ Where:

• $\tau_d$ = Design shear stress ($\text{lb/ft}^2$ or Pa)
• $\gamma$ = Unit weight of water ($62.4 \text{ lb/ft}^3$ or $9810 \text{ N/m}^3$)
• $y$ = Maximum flow depth (ft or m)
• $S$ = Channel slope (ft/ft or m/m)

Design Rule: To prevent erosion, the design shear stress ($\tau_d$) must not exceed the permissible shear stress ($\tau_p$) of the lining material: $\tau_d \le \tau_p$

If $\tau_d > \ au_p$, a more robust lining (such as larger riprap or concrete paving) must be selected.