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Hydraulics, open channels and fluid power
Specific energy lost across a hydraulic jump
Energy per unit weight dissipated across a classical hydraulic jump in a rectangular channel, from the depths y₁ before and y₂ after it, with y₂ above y₁.
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- HYD-29
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- Hydraulics, open channels and fluid power
- Result
- ΔE [m]
Across a classical hydraulic jump in a rectangular channel the specific energy falls by ΔE = (y₂ − y₁)³/(4 y₁ y₂), where y₁ and y₂ are the depths before and after. The result is the energy dissipated per unit weight of liquid, in metres. It rises sharply with the depth ratio, so strong jumps dissipate most of the approaching energy. Use y₂ above y₁.
Energy per unit weight dissipated across a classical hydraulic jump in a rectangular channel, from the depths y₁ before and y₂ after it, with y₂ above y₁.
Calculator
- Depth ratio, y₂/y₁4
- Upstream Froude number, √[(y₂/y₁)(y₂/y₁ + 1)/2]3.162
- Energy loss as a share of the upstream depth, ΔE/y₁1.688
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 jump from 0.25 m to 1 m loses 0.4219 m of head and corresponds to an upstream Froude number of 3.16.
- Jumps with Fr₁ above about 4.5 are stable and efficient, dissipating more than half of the incoming energy.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| y2 Depth after the jump | 1 to 10 m | 0.4219 | 92.69 | 92.26 | +10% | +32.3 % |
| y1 Depth before the jump | 0.01 to 0.9 m | 2.778e-4 | 24.26 | 24.26 | +10% | −17.9 % |
Sweep of Depth after the jump
| Depth after the jump (m) | Energy loss (m) |
|---|---|
| 1 | 0.4219 |
| 2.125 | 3.102 |
| 3.25 | 8.308 |
| 4.375 | 16.04 |
| 5.5 | 26.31 |
| 6.625 | 39.11 |
| 7.75 | 54.44 |
| 8.875 | 72.3 |
| 10 | 92.69 |
Sweep one variable
Variables and how each one contributes
Energy loss
What the equation gives you. Everything else feeds it.
Depth before the jump
The depth before the jump, in m. A larger y₁ reduces the depth ratio, so the energy loss changes. Raising it by 10 % (to 0.275) moves the result by −17.9 %, from 0.4219 to 0.3464 m.
Depth after the jump
The depth after the jump, in m. A larger y₂ raises the energy loss. Raising it by 10 % (to 1.1) moves the result by +32.3 %, from 0.4219 to 0.5583 m.
Worked example with the values above
- Depth change: y₂ − y₁ = 1 − 0.25 = 0.75 m; cubed 0.42188 m³.
- Denominator: 4 y₁ y₂ = 1 m².
- ΔE = 0.42188 m.
Assumptions and limits
- A prismatic rectangular channel with hydrostatic pressure and a uniform velocity profile.
- For jumps: a horizontal channel, negligible friction across the jump and momentum coefficient 1.
Common mistakes
- Using total discharge where discharge per unit width is needed.
- Applying the jump relation when Fr₁ is below 1, where no jump forms.
- Measuring the energy from the wrong datum.
Statistics and uncertainty
Depths near the critical depth are very sensitive: a small change in energy gives a large change in depth. Jump relations are also sensitive to the upstream Froude number, so errors in the measured depth and discharge matter.
Where it is used
- Locating control sections and finding whether flow is subcritical or supercritical.
- Sizing stilling basins and tracing backwater and drawdown curves.
Maths you need
- Froude number: Fr = V/√(g D), the ratio of flow speed to wave speed
- Specific energy: energy per unit weight measured from the bed
- Momentum conservation: force balance across the jump