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Hydraulics, open channels and fluid power
Mean boundary shear stress in uniform channel flow
Average shear stress over the wetted perimeter of a channel in steady, nearly uniform flow, from the hydraulic radius and the energy slope.
- Ref
- HYD-24
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- τ̄ [Pa]
In steady, nearly uniform flow the weight component along the channel balances the boundary shear. Per unit length, ρ g A S = τ̄ P, so the mean boundary shear over the wetted perimeter is τ̄ = ρ g Rh Sf. It rises with the slope and the hydraulic radius. The bed usually carries more shear than the banks, and the local value can be much higher at bends and obstacles.
Average shear stress over the wetted perimeter of a channel in steady, nearly uniform flow, from the hydraulic radius and the energy slope. Local bed and bank shear can differ from this average.
Calculator
- Shear velocity, √(τ̄/ρ)0.1085 m/s
- Mean shear if the slope doubles23.54 Pa
- Slope that would give 20 Pa0.003398
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 channel boundary carries an average shear of 11.77 Pa, a shear velocity of 0.1085 m/s.
- Fine sand starts to move at a few tenths of a pascal and coarse gravel at tens of pascals, so compare this with the critical shear of the bed material.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Pa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Sf Energy slope | 1e-5 to 0.1 | 0.05886 | 588.6 | 588.5 | +10% | +10 % |
| Rh Hydraulic radius | 0.01 to 20 m | 0.1962 | 392.4 | 392.2 | +10% | +10 % |
| g Gravitational acceleration | 1 to 30 m/s² | 1.2 | 36 | 34.8 | +10% | +10 % |
| ρ Liquid density | 500 to 2,000 kg/m³ | 5.886 | 23.54 | 17.66 | +10% | +10 % |
Sweep of Energy slope
| Energy slope | Mean boundary shear stress (Pa) |
|---|---|
| 1e-5 | 0.05886 |
| 3.162e-5 | 0.1861 |
| 1e-4 | 0.5886 |
| 3.162e-4 | 1.861 |
| 0.001 | 5.886 |
| 0.003162 | 18.61 |
| 0.01 | 58.86 |
| 0.03162 | 186.1 |
| 0.1 | 588.6 |
Sweep one variable
Variables and how each one contributes
Mean boundary shear stress
What the equation gives you. Everything else feeds it.
Liquid density
The liquid density, in kg/m³. Raises the shear in direct proportion. Raising it by 10 % (to 1,100) moves the result by +10 %, from 11.77 to 12.95 Pa.
Gravitational acceleration
The gravitational acceleration, in m/s². Raises the shear in direct proportion. Raising it by 10 % (to 10.79) moves the result by +10 %, from 11.77 to 12.95 Pa.
Hydraulic radius
The hydraulic radius, in m. Raises the shear in direct proportion. Raising it by 10 % (to 0.66) moves the result by +10 %, from 11.77 to 12.95 Pa.
Energy slope
The energy (friction) slope, a plain ratio. Raises the shear in direct proportion. Raising it by 10 % (to 0.0022) moves the result by +10 %, from 11.77 to 12.95 Pa.
Worked example with the values above
- Product: ρ g = 1,000 × 9.81 = 9,810 N/m³.
- Multiply by Rh and Sf: 0.6 × 0.002 = 0.0012 m.
- τ̄ = 11.772 Pa.
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