Open Channel Flow and Specific Energy
Uniform flow with Manning's and Chezy's equations, most economical channel sections, specific energy, critical depth, Froude number and gradually varied flow profiles.
Rivers, canals and sewers flow with a free surface open to the atmosphere. The driving force is gravity, and the depth adjusts itself to the flow.
Uniform flow
Depth, velocity and area stay constant along the channel; the bed slope equals the energy slope.
- Manning: V = (1/n) R^(2/3) S^(1/2)
- Chezy: V = C √(R S)
- Hydraulic radius R = A/P (area / wetted perimeter). For a wide rectangular channel, R ≈ y.
- Conveyance K = (1/n) A R^(2/3), so Q = K √S.
Most economical (efficient) sections
For a given area, the best section has the least wetted perimeter (so the most flow):
| Section | Condition |
|---|---|
| Rectangular | b = 2y, R = y/2 |
| Trapezoidal | Half of a regular hexagon: side slopes 60° to the horizontal, R = y/2 |
| Triangular | Vertex angle 90° (side slopes 1 : 1) |
| Circular (max discharge) | y ≈ 0.94D (max velocity at y ≈ 0.81D) |
Specific energy
Energy measured from the channel bed: E = y + V²/(2g) = y + q²/(2gy²) (q = discharge per unit width).
- For a given q, E has a minimum at the critical depth: y_c = (q²/g)^(1/3), with E_min = 1.5 y_c (rectangular channel).
- Froude number Fr = V/√(g D) (D = hydraulic depth A/T). Fr < 1 subcritical (deep, slow), Fr = 1 critical, Fr > 1 supercritical (shallow, fast).
- For other shapes, critical flow occurs when Q²T/(gA³) = 1.
Gradually varied flow
dy/dx = (S₀ − S_f)/(1 − Fr²)
Comparing the normal depth y_n and critical depth y_c classifies slopes as mild (y_n > y_c, M curves), steep (S curves), critical (C), horizontal (H) and adverse (A). The M1 profile is the backwater curve behind a dam; M2 is the drawdown ahead of a free fall.
Worked example
A rectangular channel 4 m wide carries 8 m³/s.
- q = 2 m²/s; y_c = (4/9.81)^(1/3) = 0.742 m; E_min = 1.113 m.
Practice
A wide rectangular channel carries q = 3 m²/s (g = 9.81 m/s²). What is the critical depth in m? (Give 3 decimal places.)
Show answer and explanation
Answer: 0.972y_c = (q²/g)^(1/3) = (9/9.81)^(1/3) = 0.972 m.
A rectangular channel has a bed width of 3 m and flow depth 1.5 m. What is its hydraulic radius in m?
Show answer and explanation
Answer: 0.75R = A/P = 4.5/6 = 0.75 m (this is the most economical rectangular section, b = 2y).
A channel has R = 1 m, S = 0.0004 and n = 0.02. What is the velocity by Manning's formula in m/s?
Show answer and explanation
Answer: 1V = (1/0.02) × 1 × 0.02 = 1 m/s.
For a rectangular channel, the minimum specific energy is how many times the critical depth?
Show answer and explanation
Answer: 1.5At critical flow V²/(2g) = y_c/2, so E_min = 1.5y_c.