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Hydraulics, open channels and fluid power
Diameter of a falling free liquid jet
Diameter of a vertical circular jet after it has fallen a distance H from a section of diameter d₁ and speed U₁, from continuity and Bernoulli at atmospheric pressure, with no drag, break-up or surface tension.
- Ref
- HYD-35
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- d2 [m]
A liquid jet falling vertically speeds up, and continuity makes it narrow. Bernoulli at atmospheric pressure gives the speed after a fall H as U₂ = √(U₁² + 2gH), and continuity (U₁ d₁² = U₂ d₂²) then gives d₂ = d₁[U₁²/(U₁² + 2gH)]^(1/4). The stream breaks up before the formula fails, as surface tension and air drag are neglected.
Diameter of a vertical circular jet after it has fallen a distance H from a section of diameter d₁ and speed U₁, from continuity and Bernoulli at atmospheric pressure, with no drag, break-up or surface tension.
Calculator
- Speed after the fall, √(U₁² + 2gH)3.716 m/s
- Diameter ratio, d₂/d₁0.7336
- Area ratio, (d₂/d₁)²0.5382
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.
- After 0.5 m of fall the 20 mm jet narrows to 14.67 mm, as its speed rises from 2 to 3.716 m/s.
- The slower the jet starts, the stronger the narrowing: a jet leaving at near zero speed narrows rapidly over the first few centimetres.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| d1 Initial jet diameter | 0.001 to 1 m | 7.336e-4 | 0.7336 | 0.7329 | +10% | +10 % |
| U1 Initial jet speed | 0.05 to 50 m/s | 0.002527 | 0.01998 | 0.01745 | +10% | +3.34 % |
| H Fall height | 0 to 50 m | 0.005049 | 0.02 | 0.01495 | +10% | −1.7 % |
| g Gravitational acceleration | 1 to 30 m/s² | 0.01171 | 0.01891 | 0.007202 | +10% | −1.7 % |
Sweep of Fall height
| Fall height (m) | Jet diameter after the fall (m) |
|---|---|
| 0 | 0.02 |
| 6.25 | 0.008432 |
| 12.5 | 0.007118 |
| 18.75 | 0.006441 |
| 25 | 0.005998 |
| 31.25 | 0.005675 |
| 37.5 | 0.005423 |
| 43.75 | 0.005219 |
| 50 | 0.005049 |
Sweep one variable
Variables and how each one contributes
Jet diameter after the fall
What the equation gives you. Everything else feeds it.
Initial jet diameter
The initial jet diameter, in m. Raises the final diameter in direct proportion. Raising it by 10 % (to 0.022) moves the result by +10 %, from 0.01467 to 0.01614 m.
Initial jet speed
The initial jet speed, in m/s. A faster start means less speed-up relative to the start, so the jet narrows less. Raising it by 10 % (to 2.2) moves the result by +3.34 %, from 0.01467 to 0.01516 m.
Gravitational acceleration
The gravitational acceleration, in m/s². Stronger gravity speeds the jet more, so it narrows more. Raising it by 10 % (to 10.79) moves the result by −1.7 %, from 0.01467 to 0.01442 m.
Fall height
The fall height, in m. A longer fall speeds the jet more, so it narrows more. Raising it by 10 % (to 0.55) moves the result by −1.7 %, from 0.01467 to 0.01442 m.
Worked example with the values above
- Speed after the fall: √(2² + 2 × 9.81 × 0.5) = 3.7162 m/s.
- Speed ratio U₁²/U₂² = 0.28965; fourth root = 0.73361.
- d₂ = 0.02 × 0.73361 = 0.014672 m.
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
- Using the area of the pipe where the jet area is needed.
- Mixing gauge and absolute pressures in a force balance.
- Forgetting that the force on the support is the reaction to the force on the fluid.
Statistics and uncertainty
Discharge and velocity coefficients vary with the Reynolds number and with the geometry, and measured jet distances carry the reading error of the ruler, so the results are no better than 2 % to 5 %.
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