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Hydraulics, open channels and fluid power
Velocity-area discharge by the midsection method
Discharge of a river or channel from three verticals, each with an allocated width w, depth d and depth-mean velocity u normal to the section.
- Ref
- HYD-17
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- Q [m³/s]
In the velocity–area method the cross-section is divided into verticals, each representing a width w. The flow through a panel is its area w d times the depth-mean velocity u normal to the section, and the discharge is the sum Q ≈ Σ wᵢdᵢuᵢ. This calculator uses three verticals. More verticals give a better approximation of the profile, and the end panels need a consistent convention at the banks.
Discharge of a river or channel from three verticals, each with an allocated width w, depth d and depth-mean velocity u normal to the section. The products are summed over the verticals.
Calculator
- Total width5 m
- Total flow area, Σ w d3.5 m²
- Mean velocity over the section, Q/area0.4857 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 carries 1.7 m³/s over 3.5 m² of flow area, a mean velocity of 0.4857 m/s.
- The middle panel carries the largest share (70.6 %), which is why gauging effort goes into the deepest, fastest verticals.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m³/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| w2 Width of vertical 2 | 0.1 to 100 m | 0.56 | 60.5 | 59.94 | +10% | +7.06 % |
| d2 Depth at vertical 2 | 0.01 to 20 m | 0.512 | 24.5 | 23.99 | +10% | +7.06 % |
| u2 Mean velocity at vertical 2 | 0 to 10 m/s | 0.5 | 20.5 | 20 | +10% | +7.06 % |
| w1 Width of vertical 1 | 0.1 to 100 m | 1.32 | 21.3 | 19.98 | +10% | +2.35 % |
| d1 Depth at vertical 1 | 0.01 to 20 m | 1.308 | 17.3 | 15.99 | +10% | +2.35 % |
| u1 Mean velocity at vertical 1 | 0 to 10 m/s | 1.3 | 11.3 | 10 | +10% | +2.35 % |
| w3 Width of vertical 3 | 0.1 to 100 m | 1.61 | 11.6 | 9.99 | +10% | +0.588 % |
| u3 Mean velocity at vertical 3 | 0 to 10 m/s | 1.6 | 6.6 | 5 | +10% | +0.588 % |
| d3 Depth at vertical 3 | 0.01 to 20 m | 1.602 | 5.6 | 3.998 | +10% | +0.588 % |
Sweep of Mean velocity at vertical 2
| Mean velocity at vertical 2 (m/s) | Discharge (m³/s) |
|---|---|
| 0 | 0.5 |
| 1.25 | 3 |
| 2.5 | 5.5 |
| 3.75 | 8 |
| 5 | 10.5 |
| 6.25 | 13 |
| 7.5 | 15.5 |
| 8.75 | 18 |
| 10 | 20.5 |
Sweep one variable
Variables and how each one contributes
Discharge
What the equation gives you. Everything else feeds it.
Width of vertical 1
The width allocated to vertical 1, in m. A wider first panel raises its flow in direct proportion. Raising it by 10 % (to 2.2) moves the result by +2.35 %, from 1.7 to 1.74 m³/s.
Depth at vertical 1
The depth at vertical 1, in m. A deeper first vertical raises its flow in direct proportion. Raising it by 10 % (to 0.55) moves the result by +2.35 %, from 1.7 to 1.74 m³/s.
Mean velocity at vertical 1
The mean velocity at vertical 1, in m/s. A faster first vertical raises its flow in direct proportion. Raising it by 10 % (to 0.44) moves the result by +2.35 %, from 1.7 to 1.74 m³/s.
Width of vertical 2
The width allocated to vertical 2, in m. A wider second panel raises its flow in direct proportion. Raising it by 10 % (to 2.2) moves the result by +7.06 %, from 1.7 to 1.82 m³/s.
Depth at vertical 2
The depth at vertical 2, in m. A deeper second vertical raises its flow in direct proportion. Raising it by 10 % (to 1.1) moves the result by +7.06 %, from 1.7 to 1.82 m³/s.
Mean velocity at vertical 2
The mean velocity at vertical 2, in m/s. A faster second vertical raises its flow in direct proportion. Raising it by 10 % (to 0.66) moves the result by +7.06 %, from 1.7 to 1.82 m³/s.
Width of vertical 3
The width allocated to vertical 3, in m. A wider third panel raises its flow in direct proportion. Raising it by 10 % (to 1.1) moves the result by +0.588 %, from 1.7 to 1.71 m³/s.
Depth at vertical 3
The depth at vertical 3, in m. A deeper third vertical raises its flow in direct proportion. Raising it by 10 % (to 0.55) moves the result by +0.588 %, from 1.7 to 1.71 m³/s.
Mean velocity at vertical 3
The mean velocity at vertical 3, in m/s. A faster third vertical raises its flow in direct proportion. Raising it by 10 % (to 0.22) moves the result by +0.588 %, from 1.7 to 1.71 m³/s.
Worked example with the values above
- Panel flows: 0.4, 1.2 and 0.1 m³/s.
- Sum them.
- Q = 1.7 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