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Hydraulics, open channels and fluid power
Jet velocity coefficient from its trajectory
Velocity coefficient of a small horizontal jet from its horizontal travel x, its vertical drop Y below the vena contracta, and the reservoir head h above the jet.
- Ref
- HYD-32
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- Cv
A small jet leaving a reservoir at head h has an ideal speed √(2gh), reduced by friction to Cv √(2gh). A horizontal jet falls by Y in the time x/U, so Y = g x²/(2U²). Eliminating U gives Cv = x/(2√(hY)). Measuring the jet’s landing position therefore gives the velocity coefficient. Y must be measured from the vena contracta, and drag by the air is neglected.
Velocity coefficient of a small horizontal jet from its horizontal travel x, its vertical drop Y below the vena contracta, and the reservoir head h above the jet. Air drag is neglected.
Calculator
- Ideal horizontal range, 2√(h Y)1 m
- Head lost in the nozzle, 1 − Cv², as a share of h0.0396
- Actual jet speed (g = 9.81)4.341 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 jet landed 0.98 m away, against an ideal 1 m, so Cv = 0.98.
- A value near 0.97 to 0.99 is typical for a good nozzle; a much lower value suggests friction, or a poor measurement of where the jet starts.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| x Horizontal travel | 0.01 to 10 m | 0.01 | 10 | 9.99 | +10% | +10 % |
| h Reservoir head above the jet | 0.02 to 20 m | 0.2191 | 6.93 | 6.711 | +10% | −4.65 % |
| Y Vertical drop below the vena contracta | 0.01 to 10 m | 0.155 | 4.9 | 4.745 | +10% | −4.65 % |
Sweep of Horizontal travel
| Horizontal travel (m) | Velocity coefficient |
|---|---|
| 0.01 | 0.01 |
| 0.02371 | 0.02371 |
| 0.05623 | 0.05623 |
| 0.1334 | 0.1334 |
| 0.3162 | 0.3162 |
| 0.7499 | 0.7499 |
| 1.778 | 1.778 |
| 4.217 | 4.217 |
| 10 | 10 |
Sweep one variable
Variables and how each one contributes
Velocity coefficient
What the equation gives you. Everything else feeds it.
Horizontal travel
The horizontal distance travelled by the jet, in m. A longer range means a faster jet, so Cv rises in direct proportion. Raising it by 10 % (to 1.078) moves the result by +10 %, from 0.98 to 1.078.
Reservoir head above the jet
The reservoir head above the jet, in m. A higher head means a higher ideal speed, so Cv falls for the same range. Raising it by 10 % (to 1.1) moves the result by −4.65 %, from 0.98 to 0.9344.
Vertical drop below the vena contracta
The vertical drop from the vena contracta, in m. A longer drop means a longer fall time, so Cv falls for the same range. Raising it by 10 % (to 0.275) moves the result by −4.65 %, from 0.98 to 0.9344.
Worked example with the values above
- Ideal range: 2√(h Y) = 2 × √(1 × 0.25) = 1 m.
- Ratio of actual to ideal: 0.98 / 1.
- Cv = 0.98.
Assumptions and limits
- Steady, incompressible, inviscid flow outside the boundary layer, at atmospheric pressure where stated.
- Uniform velocity profiles at the sections used.
Common mistakes
- Measuring the drop from the nozzle centre instead of from the vena contracta.
- Using the total height of the reservoir in place of the head above the jet.
- Neglecting air drag on a long jet.
Statistics and uncertainty
The coefficient depends on a small difference between measured and ideal distances, so a 1 mm error in the jet range of a 1 m jet changes Cv by about 0.1 %, and the origin of the drop is the largest uncertainty.
Where it is used
- Timing tank drainage and calibrating nozzles and orifices.
- Sizing thrust blocks, deflectors and jet impact plates.
Maths you need
- Torricelli’s law: jet speed √(2gh) from a head h
- Momentum equation: force = rate of change of momentum
- Continuity: Q = A U