Open channel flow is the flow of a liquid within a conduit with a free surface subjected to atmospheric pressure. This applies to natural rivers, engineered canals, partially full storm sewers, and spillways. The driving force is gravity, and the resisting force is boundary friction. Key concepts include Manning's Equation, Tractive Force Method, Specific Energy (and channel transitions), the Froude Number, and rapidly varied flows like the Hydraulic Jump.

Uniform flow occurs when the depth of flow, water area, and velocity remain constant along the length of the channel. This implies that the gravitational forces pushing the water downhill are exactly balanced by the frictional forces from the channel bed and banks.

The universally accepted empirical formula to analyze uniform open channel flow is Manning's Equation (historically predated by the Chezy equation, $v = C \sqrt{RS}$).

When designing unlined, erodible earth canals, simply using Manning's equation is insufficient because high velocities will scour the channel.

The channel dimensions must be sized so that $\tau_0 \le \tau_{\text{permissible}}$. Here $\gamma_w$ is the specific weight of water.

Specific energy ($E$) in an open channel is defined as the total energy head of the water measured relative to the channel bottom. It is the sum of the flow depth ($y$) and the velocity head.

If you plot Specific Energy ($E$) against depth ($y$), you obtain a curve showing that for any given specific energy greater than a minimum value, there are two possible flow depths called alternate depths.

Specific energy is critical for analyzing localized transitions like a raised channel bottom (a bump) or a narrowed width (a contraction). As flow goes over a bump, specific energy decreases. If the bump is too high, the specific energy drops to the absolute minimum (critical depth). If the bump is raised any further, a choke condition occurs: the flow must back up upstream to gain enough energy to pass over the obstruction.

The flow state relative to critical depth is classified using the Froude Number (ratio of inertial to gravity forces).

Unlike uniform flow where depth is constant, Gradually Varied Flow (GVF) occurs when depth changes slowly due to backwater effects from dams or changes in slope. Engineers calculate the continuous water surface profile. Profiles are classified based on the channel slope (Mild, Steep) and the depth relative to normal ($y_n$) and critical depth ($y_c$). Common profiles like the M1 curve (backwater behind a dam) determine levee heights.

A hydraulic jump is a rapidly varied flow phenomenon that occurs when a high-velocity, supercritical flow abruptly transitions into a slow-moving, subcritical flow, resulting in a churning surface and massive energy loss.

Engineers intentionally induce hydraulic jumps at the base of spillways to safely dissipate kinetic energy before water enters fragile natural riverbeds.

Unlike specific energy (which decreases across a jump), Specific Force (or Momentum Flux) is conserved across the jump. The depths before ($y_1$) and after ($y_2$) the jump are called conjugate depths. They are related by the momentum equation.

The energy lost ($\Delta E$) during the jump in a rectangular channel can be calculated from the depths before and after the jump.

In Water Resources Engineering, the practical application of theoretical formulas often requires careful consideration of real-world variables, such as varying friction coefficients, unpredictable environmental conditions, and changing climate patterns. A rigorous approach to empirical validation and an understanding of the safety margins involved are paramount for resilient infrastructure design.