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Hydraulics, open channels and fluid power
Critical depth of a rectangular channel
Depth that minimises specific energy for a given discharge per unit width q in a rectangular channel with hydrostatic pressure and a uniform velocity profile.
- Ref
- HYD-27
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- yc [m]
The specific energy of a rectangular channel is smallest when the Froude number is one. Differentiating E = y + q²/(2g y²) and setting dE/dy = 0 gives yc = (q²/g)^(1/3), independent of slope and roughness. At that depth the velocity head is half the depth, so Emin = 1.5 yc. The depth is a control: subcritical flows are controlled from downstream, supercritical from upstream.
Depth that minimises specific energy for a given discharge per unit width q in a rectangular channel with hydrostatic pressure and a uniform velocity profile. At this depth the Froude number is one.
Calculator
- Minimum specific energy, 3yc/21.112 m
- Critical velocity, √(g yc)2.697 m/s
- Critical depth if the flow doubles1.177 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.
- For 2 m²/s the critical depth is 0.7415 m and the critical speed is 2.697 m/s; deeper flow is subcritical, shallower is supercritical.
- The minimum energy needed to pass this flow is 1.112 m, so a step in the bed that raises the energy above that would choke the flow.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| q Discharge per unit width | 0.01 to 50 m²/s | 0.02168 | 6.34 | 6.318 | +10% | +6.56 % |
| g Gravitational acceleration | 1 to 30 m/s² | 0.5109 | 1.587 | 1.077 | +10% | −3.13 % |
Sweep of Discharge per unit width
| Discharge per unit width (m²/s) | Critical depth (m) |
|---|---|
| 0.01 | 0.02168 |
| 0.029 | 0.04409 |
| 0.08409 | 0.08966 |
| 0.2438 | 0.1823 |
| 0.7071 | 0.3708 |
| 2.05 | 0.754 |
| 5.946 | 1.533 |
| 17.24 | 3.118 |
| 50 | 6.34 |
Sweep one variable
Variables and how each one contributes
Critical depth
What the equation gives you. Everything else feeds it.
Discharge per unit width
The discharge per unit width, in m²/s. Raises the critical depth with the power 2/3. Raising it by 10 % (to 2.2) moves the result by +6.56 %, from 0.7415 to 0.7902 m.
Gravitational acceleration
The gravitational acceleration, in m/s². Lowers the critical depth with the power 1/3. Raising it by 10 % (to 10.79) moves the result by −3.13 %, from 0.7415 to 0.7183 m.
Worked example with the values above
- Square the flow: q² = 4 m⁴/s².
- Divide by g: 0.40775 m³.
- yc = (0.40775)^(1/3) = 0.74153 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