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Hydraulics, open channels and fluid power
Conjugate depth of a rectangular hydraulic jump
Depth downstream of a steady hydraulic jump in a horizontal rectangular channel, from the upstream depth y₁ and Froude number Fr₁ (above 1).
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- HYD-28
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- Hydraulics, open channels and fluid power
- Result
- y2 [m]
A hydraulic jump turns supercritical flow into subcritical flow and dissipates energy. In a horizontal rectangular channel with hydrostatic end sections, conserving momentum gives the conjugate depth y₂ = (y₁/2)[√(1 + 8Fr₁²) − 1]. It needs Fr₁ ≥ 1; at Fr₁ = 1 there is no jump (y₂ = y₁). Higher Fr₁ gives a larger depth ratio and a stronger jump.
Depth downstream of a steady hydraulic jump in a horizontal rectangular channel, from the upstream depth y₁ and Froude number Fr₁ (above 1). It conserves momentum with hydrostatic end sections and no external drag.
Calculator
- Depth ratio, y₂/y₁3.772
- Downstream Froude number, Fr₁(y₁/y₂)^1.50.4095
- Energy lost, (y₂ − y₁)³/(4y₁y₂)0.4235 m
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 0.3 m at Fr = 3 rises to 1.132 m, 3.77 times the initial depth, and leaves a flow with Fr = 0.41.
- The jump loses 0.4235 m of head, which the stilling basin must be able to withstand.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| y1 Depth before the jump | 0.01 to 5 m | 0.03772 | 18.86 | 18.82 | +10% | +10 % |
| Fr1 Froude number before the jump | 1 to 20 | 0.3 | 8.337 | 8.037 | +10% | +11.2 % |
Sweep of Froude number before the jump
| Froude number before the jump | Conjugate depth after the jump (m) |
|---|---|
| 1 | 0.3 |
| 3.375 | 1.29 |
| 5.75 | 2.294 |
| 8.125 | 3.3 |
| 10.5 | 4.307 |
| 12.88 | 5.314 |
| 15.25 | 6.322 |
| 17.63 | 7.329 |
| 20 | 8.337 |
Sweep one variable
Variables and how each one contributes
Conjugate depth after the jump
What the equation gives you. Everything else feeds it.
Depth before the jump
The depth upstream of the jump, in m. Raises the downstream depth in direct proportion. Raising it by 10 % (to 0.33) moves the result by +10 %, from 1.132 to 1.245 m.
Froude number before the jump
The Froude number upstream of the jump (1 or more). A higher Froude number gives a larger conjugate depth. Raising it by 10 % (to 3.3) moves the result by +11.2 %, from 1.132 to 1.258 m.
Worked example with the values above
- Square root: √(1 + 8 × 3²) = √73 = 8.544.
- Subtract 1 and multiply by y₁/2: 0.15 × 7.544.
- y₂ = 1.1316 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