12.2 Energy and Momentum Principles

Energy and Momentum Principles in Open Channel Flows

Open channel flows are governed by energy and momentum principles. These concepts help engineers analyze and predict flow behavior in rivers, canals, and other open waterways. Understanding energy equations, critical depth, and specific energy is crucial for designing hydraulic structures and managing water resources effectively.

Momentum principles play a vital role in analyzing hydraulic jumps, which occur when flow transitions from supercritical to subcritical. These principles help engineers design stilling basins and energy dissipators to control flow and prevent erosion downstream of hydraulic structures like dams.

Energy Equation in Open Channels

The energy equation for open channels comes from the Bernoulli equation. At any cross-section, total energy is the sum of three components:

• Elevation head ($z$): the height of the channel bottom above a reference datum
• Pressure head ($\frac{p}{\gamma}$): related to the fluid pressure at that point
• Velocity head ($\frac{V^2}{2g}$): the kinetic energy per unit weight of the fluid

Between two cross-sections, the full energy equation is:

$z_1 + \frac{p_1}{\gamma} + \frac{V_1^2}{2g} + h_p = z_2 + \frac{p_2}{\gamma} + \frac{V_2^2}{2g} + h_L$

where $h_p$ is energy added by a pump (if present) and $h_L$ is energy lost to friction and other factors.

In open channels, the pressure distribution is hydrostatic, so the pressure head at any point simply equals the depth of flow $y$ above that point. This lets you replace $\frac{p}{\gamma}$ with $y$, giving the simplified open-channel energy equation:

$z_1 + y_1 + \frac{V_1^2}{2g} = z_2 + y_2 + \frac{V_2^2}{2g} + h_L$

This form is what you'll use most often. It tracks how flow depth, velocity, and energy change between cross-sections along a channel, such as upstream and downstream of culverts or weirs.

Critical Depth: Concept and Significance

Critical depth ($y_c$) is the flow depth at which specific energy reaches its minimum for a given discharge. It marks the boundary between two fundamentally different flow regimes, identified by the Froude number:

$Fr = \frac{V}{\sqrt{gy}}$

where $V$ is the average velocity, $g$ is gravitational acceleration, and $y$ is flow depth.

• $Fr < 1$: Subcritical (tranquil) flow. Gravity forces dominate. The flow is relatively deep and slow. Disturbances can travel upstream.
• $Fr = 1$: Critical flow. This is the transition point, occurring at critical depth.
• $Fr > 1$: Supercritical (rapid) flow. Inertial forces dominate. The flow is shallow and fast. Disturbances cannot travel upstream.

At critical depth, flow is inherently unstable. Small perturbations can push it into either regime, which is why hydraulic jumps tend to form near critical conditions.

For a rectangular channel, critical depth is calculated as:

$y_c = \sqrt[3]{\frac{q^2}{g}}$

where $q = \frac{Q}{b}$ is the discharge per unit width, $Q$ is total discharge, and $b$ is channel width. This equation applies specifically to rectangular cross-sections; other geometries require solving a more general relationship between area, top width, and discharge.

Critical depth is central to the design of spillways, chutes, and transitions where the flow regime must be carefully controlled.

Specific Energy and Critical Depth

Specific energy ($E$) is the total energy measured relative to the channel bottom (rather than a fixed datum). It combines flow depth and velocity head:

$E = y + \frac{V^2}{2g}$

Since $V = \frac{Q}{A}$, you can also write this in terms of discharge and cross-sectional area, which makes it clear that for a fixed discharge, $E$ depends only on $y$.

The specific energy diagram plots flow depth $y$ (vertical axis) against specific energy $E$ (horizontal axis) for a given discharge. The curve has two branches:

• The upper branch corresponds to subcritical flow (deeper, slower).
• The lower branch corresponds to supercritical flow (shallower, faster).

The two branches meet at the critical point, where $E$ is minimized. To find critical depth analytically, set the derivative of $E$ with respect to $y$ equal to zero. For a rectangular channel, this condition simplifies to:

$\frac{dE}{dy} = 1 - \frac{q^2}{gy^3} = 0$

Solving this gives the same critical depth formula from the previous section. The physical meaning: at critical depth, the rate at which potential energy decreases (as depth drops) exactly equals the rate at which kinetic energy increases (as velocity rises).

Typical problems you'll encounter:

1. Determining critical depth for a given discharge and channel geometry
2. Finding the discharge needed to maintain a specific flow depth
3. Analyzing how specific energy and flow depth change when channel slope or roughness changes (e.g., transitioning from a concrete-lined channel to a natural earth channel)

Momentum Equation for Hydraulic Jumps

While the energy equation tracks energy conservation (with losses), the momentum equation applies Newton's second law to a control volume of fluid. This is especially useful for hydraulic jumps, where energy losses are large and difficult to calculate directly.

The general momentum equation between two cross-sections is:

$\beta_1 \rho Q V_1 + \frac{\gamma}{2} A_1 \bar{y}_1 = \beta_2 \rho Q V_2 + \frac{\gamma}{2} A_2 \bar{y}_2 + F_f$

where:

• $\beta$ is the momentum correction factor, accounting for non-uniform velocity distribution across the section ($\beta = 1$ for uniform velocity)
• $\rho$ is fluid density
• $Q$ is discharge
• $A$ is cross-sectional area of flow
• $\bar{y}$ is the depth from the water surface to the centroid of the cross-sectional area
• $F_f$ represents external forces such as friction and pressure forces from channel boundaries

A hydraulic jump occurs when supercritical flow is forced to transition to subcritical flow. The result is a sudden, turbulent increase in depth and a corresponding decrease in velocity. Significant energy is dissipated as turbulence and heat during this process.

The depths upstream ($y_1$) and downstream ($y_2$) of a hydraulic jump are called conjugate depths. For a rectangular channel with $\beta = 1$ and neglecting friction ($F_f \approx 0$), the conjugate depth relationship is:

$\frac{y_2}{y_1} = \frac{1}{2}\left(-1 + \sqrt{1 + 8Fr_1^2}\right)$

This is one of the most useful equations in open channel flow. It connects the upstream Froude number directly to the depth ratio across the jump.

The upstream Froude number ($Fr_1$) also classifies the type of hydraulic jump:

Engineers use these classifications when designing stilling basins and energy dissipators downstream of dams and spillways. The goal is to force a controlled hydraulic jump that dissipates excess kinetic energy before the flow continues downstream, preventing scour and erosion of the channel bed.