Library / Hydraulics, open channels and fluid power / HYD-22
Hydraulics, open channels and fluid power
Hydraulic depth of a trapezoidal channel
Flow area divided by free-surface top width for a trapezoidal channel.
- Ref
- HYD-22
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- Dh [m]
The hydraulic depth is the flow area divided by the top width, Dh = A/T = y(b + zy)/(b + 2zy) for a trapezoid. It is the depth a rectangular channel of the same top width would have for the same area. It enters the Froude number, Fr = V/√(g Dh), and the long-wave speed √(g Dh). Do not confuse it with the hydraulic radius, which uses the wetted perimeter.
Flow area divided by free-surface top width for a trapezoidal channel. It is the depth used in the Froude number and in long-wave and critical-flow calculations.
Calculator
- Flow area, y(b + zy)4.75 m²
- Top width, b + 2zy5.5 m
- Long-wave speed, √(9.81 Dh)2.911 m/s
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.
- The section has a hydraulic depth of 0.8636 m, less than its 1 m depth because the banks widen the surface.
- Waves travel at about 2.911 m/s on this section, and flow is critical when the water moves at that speed.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| y Flow depth | 0.05 to 10 m | 0.04954 | 6.053 | 6.003 | +10% | +8.77 % |
| b Bed width | 0 to 100 m | 0.5 | 0.9926 | 0.4926 | +10% | +1.07 % |
| z Side slope (horizontal per unit vertical) | 0 to 5 | 0.6429 | 1 | 0.3571 | +10% | −1.12 % |
Sweep of Side slope (horizontal per unit vertical)
| Side slope (horizontal per unit vertical) | Hydraulic depth (m) |
|---|---|
| 0 | 1 |
| 0.625 | 0.881 |
| 1.25 | 0.8077 |
| 1.875 | 0.7581 |
| 2.5 | 0.7222 |
| 3.125 | 0.6951 |
| 3.75 | 0.6739 |
| 4.375 | 0.6569 |
| 5 | 0.6429 |
Sweep one variable
Variables and how each one contributes
Hydraulic depth
What the equation gives you. Everything else feeds it.
Bed width
The bed width, in m. A wider bed brings the hydraulic depth closer to the flow depth. Raising it by 10 % (to 4.4) moves the result by +1.07 %, from 0.8636 to 0.8729 m.
Flow depth
The flow depth, in m. A greater depth raises the hydraulic depth. Raising it by 10 % (to 1.1) moves the result by +8.77 %, from 0.8636 to 0.9394 m.
Side slope (horizontal per unit vertical)
The side slope: horizontal run per unit vertical rise on each bank. Flatter banks widen the top, lowering the hydraulic depth. Raising it by 10 % (to 0.825) moves the result by −1.12 %, from 0.8636 to 0.854 m.
Worked example with the values above
- Area: 1 × (4 + 0.75 × 1) = 4.75 m².
- Top width: 4 + 2 × 0.75 × 1 = 5.5 m.
- Dh = 4.75 / 5.5 = 0.86364 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