Library / Hydraulics, open channels and fluid power / HYD-19
Hydraulics, open channels and fluid power
Triangular sharp-crested weir discharge
Free discharge over a V-notch weir of full notch angle θ (degrees) under head H above the vertex, with a ventilated nappe, negligible approach velocity and a calibrated coefficient.
- Ref
- HYD-19
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- Q [m³/s]
For a V-notch weir of full angle θ the width of the nappe grows linearly with head, 2H tan(θ/2), and integrating gives Q = (8/15) Cd tan(θ/2) √(2g) H^(5/2). The higher power of H means a V-notch gives a larger head change for the same flow change than a rectangular weir, so it measures small flows more accurately. θ is in degrees. Standard installations need an effective-head correction.
Free discharge over a V-notch weir of full notch angle θ (degrees) under head H above the vertex, with a ventilated nappe, negligible approach velocity and a calibrated coefficient.
Calculator
- Water-surface width at the notch, 2H tan(θ/2)0.4 m
- Discharge in litres per second25.36 L/s
- Discharge if the head doubles0.1434 m³/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.
- A 90° notch with 20 cm of head passes 0.02536 m³/s (25.36 L/s).
- The head follows Q^(2/5), so a flow ten times smaller still gives a head that is 40 % of the original, which is easy to read.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m³/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| H Head above the vertex | 0.01 to 1 m | 1.417e-5 | 1.417 | 1.417 | +10% | +26.9 % |
| θ Full notch angle | 10 to 150 ° | 0.002218 | 0.09463 | 0.09241 | +10% | +17.1 % |
| g Gravitational acceleration | 1 to 30 m/s² | 0.008095 | 0.04434 | 0.03625 | +10% | +4.88 % |
| Cd Discharge coefficient | 0.4 to 0.9 | 0.0169 | 0.03803 | 0.02113 | +10% | +10 % |
Sweep of Head above the vertex
| Head above the vertex (m) | Discharge (m³/s) |
|---|---|
| 0.01 | 1.417e-5 |
| 0.1338 | 0.009273 |
| 0.2575 | 0.04769 |
| 0.3812 | 0.1272 |
| 0.505 | 0.2569 |
| 0.6288 | 0.4443 |
| 0.7525 | 0.6963 |
| 0.8762 | 1.019 |
| 1 | 1.417 |
Sweep one variable
Variables and how each one contributes
Discharge
What the equation gives you. Everything else feeds it.
Discharge coefficient
The discharge coefficient, usually about 0.6. Raises the flow in direct proportion. Raising it by 10 % (to 0.66) moves the result by +10 %, from 0.02536 to 0.02789 m³/s.
Full notch angle
The full notch angle, in degrees. A wider notch raises the flow through tan(θ/2). Raising it by 10 % (to 99) moves the result by +17.1 %, from 0.02536 to 0.02969 m³/s.
Gravitational acceleration
The gravitational acceleration, in m/s². Raises the flow with the square root of g. Raising it by 10 % (to 10.79) moves the result by +4.88 %, from 0.02536 to 0.02659 m³/s.
Head above the vertex
The head above the notch vertex, in m. Raises the flow with the power 5/2. Raising it by 10 % (to 0.22) moves the result by +26.9 %, from 0.02536 to 0.03218 m³/s.
Worked example with the values above
- Notch term: tan(θ/2) = tan(45°) = 1.
- Head term: H^(5/2) = 0.017889; √(2g) = 4.4294.
- Q = (8/15) × 0.6 × 1 × 4.4294 × 0.017889 = 0.025356 m³/s.
Assumptions and limits
- Steady flow with the discharge coefficient suited to the installation and the Reynolds number.
- For weirs: free discharge with an aerated nappe, a thin or broad crest as stated and negligible approach velocity.
Common mistakes
- Measuring the head at the crest of the weir instead of upstream, beyond the drawdown.
- Using a coefficient from a different weir type or scale.
- Using the total head where the static head over the crest is needed, or the reverse.
Statistics and uncertainty
Discharge coefficients are calibrated values, normally good to 1 % to 5 %, and the head enters with a power of 1.5 to 2.5, so a 1 % error in head gives an error in flow of 1.5 % to 2.5 %.
Where it is used
- Measuring flow in pipes, canals, rivers and flumes.
- Setting weir crest levels and sizing overflows.
Maths you need
- Discharge coefficient: the ratio of actual to ideal flow
- Bernoulli equation: energy conservation along a streamline
- Critical flow: flow at its minimum specific energy, with Froude number 1