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Hydraulics, open channels and fluid power
Normal depth of a wide rectangular channel
Uniform-flow depth in a rectangular channel much wider than it is deep, from Manning’s n, discharge Q, width b and bed slope S₀.
- Ref
- HYD-23
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- yn [m]
For steady uniform flow in a channel much wider than it is deep, the hydraulic radius is close to the depth, and Manning’s equation, q = (1/n) y^(5/3) S₀^(1/2) per unit width, gives yn ≈ [n Q/(b √S₀)]^(3/5). It is in SI units, with n in s/m^(1/3). The slope S₀ is the bed slope, equal to the energy slope in uniform flow.
Uniform-flow depth in a rectangular channel much wider than it is deep, from Manning’s n, discharge Q, width b and bed slope S₀. The hydraulic radius is approximated by the depth, in SI units.
Calculator
- Mean velocity, Q/(b yn)0.6049 m/s
- Froude number at normal depth (g = 9.81)0.3358
- Normal depth if the flow doubles0.5012 m
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- 4 m³/s in a 20 m wide channel settles at 0.3307 m deep with a velocity of 0.6049 m/s.
- The Froude number is 0.336, so the flow is subcritical (tranquil); doubling the flow raises the depth by only 51.6 %.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Q Discharge | 0.01 to 10,000 m³/s | 0.009081 | 36.15 | 36.14 | +10% | +5.89 % |
| b Channel width | 0.5 to 2,000 m | 0.02086 | 3.024 | 3.003 | +10% | −5.56 % |
| S0 Bed slope | 1e-5 to 0.1 | 0.08306 | 1.316 | 1.233 | +10% | −2.82 % |
| n Manning roughness | 0.008 to 0.2 s/m^(1/3) | 0.1669 | 1.151 | 0.9845 | +10% | +5.89 % |
Sweep of Manning roughness
| Manning roughness (s/m^(1/3)) | Normal depth (m) |
|---|---|
| 0.008 | 0.1669 |
| 0.032 | 0.3834 |
| 0.056 | 0.5365 |
| 0.08 | 0.6645 |
| 0.104 | 0.7777 |
| 0.128 | 0.8809 |
| 0.152 | 0.9766 |
| 0.176 | 1.066 |
| 0.2 | 1.151 |
Sweep one variable
Variables and how each one contributes
Normal depth
What the equation gives you. Everything else feeds it.
Manning roughness
The Manning roughness, in s/m^(1/3). A rougher channel needs a deeper flow, with a power of 3/5. Raising it by 10 % (to 0.0275) moves the result by +5.89 %, from 0.3307 to 0.3501 m.
Discharge
The discharge, in m³/s. A larger flow needs a deeper flow, with a power of 3/5. Raising it by 10 % (to 4.4) moves the result by +5.89 %, from 0.3307 to 0.3501 m.
Channel width
The channel width, in m. A wider channel spreads the flow, so the depth falls. Raising it by 10 % (to 22) moves the result by −5.56 %, from 0.3307 to 0.3123 m.
Bed slope
The bed slope, a plain ratio (m/m). A steeper bed gives a faster, shallower flow. Raising it by 10 % (to 0.0011) moves the result by −2.82 %, from 0.3307 to 0.3213 m.
Worked example with the values above
- Group: n Q/(b √S₀) = 0.025 × 4 / (20 × 0.031623) = 0.15811.
- Raise to the power 3/5 (0.6).
- yn = 0.33066 m.
Assumptions and limits
- Prismatic channels with a steady, nearly uniform flow, in SI units.
- A free surface at atmospheric pressure and a roughness that is the same all round the wetted perimeter.
Common mistakes
- Including the free surface in the wetted perimeter.
- Using the Manning formula with n in US units.
- Using the wide-channel result for a narrow channel, where the hydraulic radius is well below the depth.
Statistics and uncertainty
Manning’s n is the main source of uncertainty and is commonly known to within 10 % to 30 %. Depth and flow follow a power of 3/5 in the normal-depth formula, so a 10 % error in n gives about a 6 % error in depth.
Where it is used
- Designing canals, drains, culverts and flumes.
- Estimating flood depth and checking bed shear for scour.
Maths you need
- Wetted perimeter: the length of channel boundary in contact with the water
- Manning’s equation: V = R^(2/3) S^(1/2) / n
- Hydraulic radius: area divided by wetted perimeter