Rapidly Varied Flow

Why it matters

Rapidly Varied Flow (RVF) is crucial in hydraulic engineering as it involves sudden changes in flow depth and velocity, which are common in structures like spillways, weirs, and sluice gates. Understanding RVF helps in designing these structures to manage water flow efficiently and prevent structural damage.

Key ideas

• Definition: Rapidly Varied Flow occurs when there is a sudden change in flow conditions, typically over a short distance. This is in contrast to Gradually Varied Flow, where changes occur over a longer distance.
• Hydraulic Jump: A common example of RVF, where supercritical flow transitions to subcritical flow, resulting in a sudden rise in water surface.
• Energy Considerations: Hydraulic jumps have substantial energy loss; other rapidly varied flows need their own loss and pressure-distribution models.
• Momentum Principle: The momentum equation is often used to analyze RVF, especially in hydraulic jumps, where the change in momentum is significant.

Formulas

• E1 = E2 + hL
  • E1: Energy at section 1 (m)
  • E2: Energy at section 2 (m)
  • hL: Head loss due to turbulence (m)
• For a steady horizontal rectangular jump with negligible external shear, the momentum function per unit width is conserved: M = y²/2 + q²/(gy), where q = Q/b (m²/s), y is depth (m), and M has units m². This is not a statement that every physical force at the two sections is equal.

Worked example

Given: A horizontal rectangular channel with negligible bed shear over a steady jump and hydrostatic pressure at its end sections with a width of 3 m carries a flow of 6 m³/s. The depth of flow before the hydraulic jump is 0.5 m. Calculate the depth after the jump.

1. Calculate the velocity before the jump:

   • Q = A1 * V1
   • A1 = b * y1 = 3 * 0.5 = 1.5 m²
   • V1 = Q / A1 = 6 / 1.5 = 4 m/s

2. Apply the momentum equation:

   • y2 = (y1/2) * (sqrt(1 + 8*Fr1²) - 1)
   • Fr1 = V1 / sqrt(g * y1) = 4 / sqrt(9.81 * 0.5) = 1.8061
   • y2 = (0.5/2) × [sqrt(1 + 8 × 4²/(9.81 × 0.5)) − 1] = 1.05134 m

Final Answer: The depth after the jump is 1.051 m.

Common mistakes

• Confusing rapidly varied flow with gradually varied flow.
• Ignoring energy losses due to turbulence.
• Incorrectly applying the momentum equation without considering the flow regime.

For GATE CE

Questions on RVF often involve calculating flow depths before and after hydraulic jumps, analyzing energy losses, and applying the momentum equation. Practice problems involving different channel shapes and flow conditions.

Quick check

1. What is a hydraulic jump?
2. How does rapidly varied flow differ from gradually varied flow?
3. What principle is primarily used to analyze rapidly varied flow?

Answers: 1. A sudden rise in water surface when supercritical flow transitions to subcritical flow. 2. Rapidly varied flow involves sudden changes over a short distance, while gradually varied flow changes occur over a longer distance. 3. The momentum principle.