Gradually Varied Flow

Why it matters

Gradually Varied Flow (GVF) is essential in the design and analysis of open channels, such as rivers and canals, where the water surface profile changes gradually. Understanding GVF helps engineers predict water levels and design efficient drainage systems, ensuring safety and functionality in hydraulic structures.

Key ideas

• Gradually Varied Flow (GVF): A type of non-uniform flow where the depth of flow changes gradually along the length of the channel.
• Water Surface Profiles: Classified based on the slope of the channel bed (mild, steep, critical, horizontal, adverse) and the flow depth relative to critical and normal depths.
• Dynamic Equation of GVF: Derived from the energy equation and continuity equation, considering the balance of gravitational, frictional, and pressure forces.
• Classification of Profiles: M1, M2, M3, S1, S2, S3, C1, C3, H2, H3, A2 and A3, where 'M' stands for mild slope, 'S' for steep slope, 'C' for critical slope, 'H' for horizontal slope, and 'A' for adverse slope.

Formulas

For steady, gradually varied flow in a prismatic small-slope channel, with hydrostatic pressure and velocity-distribution coefficient taken as 1; x increases downstream. This equation does not describe a hydraulic jump.

• dy/dx = (S0 - Sf) / (1 - (Q^2 * T / (g * A^3)))
  • dy/dx: Rate of change of flow depth with respect to distance (dimensionless)
  • S0: Channel bed slope (dimensionless)
  • Sf: Friction slope (dimensionless)
  • Q: Discharge (m³/s)
  • g: Acceleration due to gravity (9.81 m/s²)
  • A: Cross-sectional area of flow (m²)
  • T: Top width of the flow (m)

Worked example

Given:

• Discharge, Q = 20 m³/s
• Channel bed slope, S0 = 0.001
• Manning's roughness coefficient, n = 0.015
• Channel width, b = 10 m
• Flow depth, y = 2 m

1. Calculate the cross-sectional area, A: A = b * y = 10 m * 2 m = 20 m²

2. Calculate the hydraulic radius, R: R = A / P = 20 m² / (10 m + 2 * 2 m) = 1.42857 m

3. Calculate the friction slope, Sf: Sf = (n^2 * Q^2) / (A^2 * R^(4/3)) = (0.015^2 * 20^2) / (20^2 * 1.42857^(4/3)) = 0.000139845

4. Calculate dy/dx: dy/dx = (S0 - Sf) / (1 - (Q^2 * T / (g * A^3))) dy/dx = (0.001 - 0.000139845) / (1 - 20^2 * 10 / (9.81 * 20^3)) = 0.00090635

Final Answer: The rate of change of flow depth, dy/dx ≈ 0.00090635 (dimensionless).

Common mistakes

• Confusing the classification of water surface profiles.
• Incorrect calculation of the friction slope, Sf.
• Neglecting the effect of channel shape on hydraulic radius.

For GATE CE

Questions often involve calculating the water surface profile, determining the type of flow profile, and applying the dynamic equation of GVF. Practice problems on identifying profile types and solving the GVF equation for different channel conditions.

Quick check

1. What is the significance of the friction slope in GVF?
2. How does the channel slope affect the classification of water surface profiles?
3. What is the role of the hydraulic radius in calculating the friction slope?

Answers: 1. It represents the energy loss due to friction. 2. It determines the type of profile (mild, steep, etc.). 3. It affects the calculation of flow resistance and energy loss.