Open Channel Hydraulics and Flow Equations

Introduction

Open channel flow describes fluid flow in a conduit with a free surface exposed to atmospheric pressure, such as rivers, canals, aqueducts, and sewers. Gravity is the primary driving force, while boundary friction resists movement.

Manning’s Equation for Uniform Flow

Under steady, uniform flow conditions, the average velocity ($v$) in an open channel is calculated using the empirical Manning’s equation:

$$v = rac{k}{n} cdot R_h^{2/3} cdot S^{1/2}$$

Where:

• $k$ is a conversion factor ($1.0$ for metric SI units, $1.486$ for US customary units).
• $n$ is Manning’s roughness coefficient (dimensionless, depending on channel surface material).
• $R_h$ is the hydraulic radius ($A/P$), where $A$ is cross-sectional flow area and $P$ is the wetted perimeter.
• $S$ is the energy slope (equal to channel bed slope under uniform flow).

Specific Energy and Critical Flow

Specific energy ($E$) is the energy of flow per unit weight of water, measured relative to the channel bottom:

$$E = y + rac{v^2}{2g}$$

Where $y$ is flow depth, and $v^2/2g$ is velocity head. For a given discharge, there is a minimum specific energy depth known as the critical depth ($y_c$).

The Froude Number

The state of flow is classified using the dimensionless Froude Number ($Fr$):

$$Fr = rac{v}{sqrt{gD}}$$

Where $D$ is the hydraulic depth ($A/T$, where $T$ is top channel width). Flow regimes are defined as:

• $Fr < 1$: Subcritical flow (deep, slow, tranquil flow).
• $Fr = 1$: Critical flow (minimum specific energy).
• $Fr > 1$: Supercritical flow (shallow, rapid, shooting flow).