12.2 Energy and Momentum Principles

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 and Momentum Principles in Open Channel Flows

Energy equation in open channels

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• Derived from Bernoulli equation accounts for elevation head ($z$), pressure head ($\frac{p}{\gamma}$), and velocity head ($\frac{V^2}{2g}$)
• Total energy at any cross-section is the sum of these three components
• Energy equation between two cross-sections: $z_1 + \frac{p_1}{\gamma} + \frac{V_1^2}{2g} + h_p = z_2 + \frac{p_2}{\gamma} + \frac{V_2^2}{2g} + h_L$
  • $h_p$ represents energy added by a pump (if present)
  • $h_L$ represents energy lost due to friction and other factors
• In open channels, pressure head equals depth of flow ($y$) due to hydrostatic pressure distribution
• Simplified energy equation for open channels: $z_1 + y_1 + \frac{V_1^2}{2g} = z_2 + y_2 + \frac{V_2^2}{2g} + h_L$
• Analyzes changes in flow depth, velocity, and energy between different cross-sections along the channel (culverts, weirs)

Critical depth concept and significance

• Depth of flow at which specific energy is minimum for a given discharge
• Represents condition where Froude number ($Fr$) equals 1
  • $Fr = \frac{V}{\sqrt{gy}}$, where $V$ is average velocity, $g$ is gravitational acceleration, and $y$ is flow depth
  • $Fr < 1$ indicates subcritical (tranquil) flow
  • $Fr > 1$ indicates supercritical (rapid) flow
• At critical depth, flow is unstable and can transition between subcritical and supercritical states (hydraulic jump)
• Critical depth equation: $y_c = \sqrt[3]{\frac{q^2}{g}}$, where $q$ is discharge per unit width ($\frac{Q}{b}$), $Q$ is total discharge, and $b$ is channel width
• Important in designing hydraulic structures (spillways, chutes) to ensure proper flow conditions

Specific energy and critical depth

• Specific energy ($E$) is the sum of flow depth and velocity head at a given cross-section: $E = y + \frac{V^2}{2g}$
• For a given discharge and channel geometry, specific energy curve plots $y$ on x-axis and $E$ on y-axis
• Critical depth corresponds to point on specific energy curve where slope is zero (minimum specific energy)
• To find critical depth, set derivative of specific energy equation to zero and solve for $y$: $\frac{dE}{dy} = 1 - \frac{V^2}{gy} \frac{dV}{dy} = 0$
• Problems involve:
  1. Determining critical depth for given discharge and channel geometry
  2. Finding discharge required to maintain specific flow depth
  3. Analyzing change in specific energy and flow depth when channel slope or roughness changes (concrete vs. natural channels)

Momentum equation for hydraulic jumps

• Based on conservation of momentum principle
• Momentum equation between two cross-sections: $\beta_1 \rho Q V_1 + \frac{\gamma}{2} A_1 y_1 = \beta_2 \rho Q V_2 + \frac{\gamma}{2} A_2 y_2 + F_f$
  • $\beta$ is momentum correction factor for non-uniform velocity distribution
  • $\rho$ is fluid density
  • $A$ is cross-sectional area of flow
  • $F_f$ represents external forces (friction, pressure)
• Hydraulic jump occurs when flow transitions from supercritical to subcritical, causing sudden increase in flow depth and decrease in velocity
• Analyzes hydraulic jump characteristics, such as conjugate depths ($y_1$ and $y_2$) and energy dissipation
• Froude number upstream of hydraulic jump ($Fr_1$) classifies jump type:
  1. $1 < Fr_1 < 1.7$: Undular jump
  2. $1.7 < Fr_1 < 2.5$: Weak jump
  3. $2.5 < Fr_1 < 4.5$: Oscillating jump
  4. $4.5 < Fr_1 < 9.0$: Steady jump
  5. $Fr_1 > 9.0$: Strong jump
• Applied to design stilling basins and energy dissipators downstream of hydraulic structures (dams) to control hydraulic jump and prevent erosion