12.4 Hydraulic Jumps and Rapidly Varied Flow

Hydraulic jumps occur when fast, shallow water suddenly slows down and gets deeper. This dramatic change causes turbulence and energy loss. The jump's intensity depends on the upstream flow speed, measured by the Froude number.

Rapidly varied flow happens when water depth changes quickly over a short distance. This occurs in structures like spillways and weirs. Understanding these flows is crucial for designing safe and efficient hydraulic structures in water management systems.

Hydraulic Jumps

Formation of hydraulic jumps

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• Hydraulic jumps form when supercritical flow transitions to subcritical flow
  • Supercritical flow has a Froude number ($Fr$) greater than 1, characterized by high velocity and low depth
  • Subcritical flow has a Froude number ($Fr$) less than 1, characterized by low velocity and high depth
• Key characteristics of hydraulic jumps include:
  • Sudden increase in water surface elevation (height of the jump)
  • Substantial energy dissipation caused by intense turbulence and mixing within the jump
  • Development of a roller (a recirculating flow region) and eddy currents near the surface
• Formation of hydraulic jumps influenced by factors such as:
  • Channel slope (steeper slopes promote supercritical flow)
  • Upstream and downstream flow conditions (discharge and depth)
  • Channel geometry (width, roughness, and shape)

Classification by Froude numbers

• Hydraulic jumps classified based on the upstream Froude number ($Fr_1$):
  • Undular jump: 1 < $Fr_1$ < 1.7
    • Characterized by a smooth, wavy water surface profile
    • Minimal energy dissipation occurs in this type of jump
  • Weak jump: 1.7 < $Fr_1$ < 2.5
    • Features a small roller and low energy dissipation
    • Relatively stable and well-defined jump
  • Oscillating jump: 2.5 < $Fr_1$ < 4.5
    • Exhibits an unstable, oscillating water surface
    • Formation of waves and eddies within the jump
  • Steady jump: 4.5 < $Fr_1$ < 9.0
    • Most stable and well-defined type of jump
    • Significant energy dissipation occurs due to turbulence
  • Strong jump: $Fr_1$ > 9.0
    • Characterized by intense turbulence and substantial energy dissipation
    • Formation of large rollers and eddies within the jump

Energy dissipation in hydraulic jumps

• Energy dissipation in hydraulic jumps results from:
  • Turbulence and mixing within the jump
  • Quantified by the head loss ($\Delta H$) across the jump
  • Calculated using the Belanger equation: $\frac{\Delta H}{y_1} = \frac{(y_2 - y_1)^3}{4y_1y_2}$
    • $y_1$ represents the upstream depth (supercritical flow)
    • $y_2$ represents the downstream depth (subcritical flow)
• Depth changes across hydraulic jumps:
  • Relationship between upstream and downstream depths given by the sequent depth equation: $\frac{y_2}{y_1} = \frac{1}{2}(\sqrt{1 + 8Fr_1^2} - 1)$
  • Conjugate depths ($y_1$ and $y_2$) form a unique pair for a given upstream Froude number ($Fr_1$)
  • Depth ratio ($y_2/y_1$) increases with increasing $Fr_1$

Rapidly Varied Flow

Flow over spillways and weirs

• Rapidly varied flow occurs when flow depth changes significantly over a short distance, such as:
  • Flow over spillways (ogee spillways, broad-crested weirs)
  • Flow over weirs (sharp-crested weirs, broad-crested weirs)
  • Flow through hydraulic structures (sluice gates, culverts)
• Specific energy concept:
  • Specific energy ($E$) is the sum of the depth ($y$) and the velocity head ($\frac{v^2}{2g}$)
  • Minimum specific energy corresponds to the critical depth ($y_c$)
  • Flow transitions (subcritical to supercritical or vice versa) occur at critical depth
• Critical flow conditions:
  • Froude number ($Fr$) equals 1 at critical flow
  • Critical depth ($y_c$) calculated as: $y_c = \sqrt[3]{\frac{q^2}{g}}$
    • $q$ is the discharge per unit width
    • $g$ is the gravitational acceleration
• Flow over spillways:
  • Broad-crested weirs: $Q = C_dL\sqrt{2g}H^{3/2}$
    • $Q$ is the discharge over the weir
    • $C_d$ is the discharge coefficient (depends on weir geometry)
    • $L$ is the length of the weir crest
    • $H$ is the upstream head above the weir crest
  • Ogee spillways designed to conform to the shape of the lower nappe of a ventilated nappe, minimizing energy loss
• Flow over sharp-crested weirs:
  • Rectangular weir: $Q = \frac{2}{3}C_d\sqrt{2g}LH^{3/2}$
    • $C_d$ is the discharge coefficient (typically 0.6-0.8)
  • V-notch (triangular) weir: $Q = \frac{8}{15}C_d\sqrt{2g}\tan(\frac{\theta}{2})H^{5/2}$
    • $\theta$ is the notch angle (e.g., 90°, 60°, 45°)