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Hydraulics, open channels and fluid power
Gradually varied flow depth gradient
Rate of change of depth along a prismatic channel with the bed slope S₀, friction slope Sf and local Froude number Fr, with x increasing downstream and hydrostatic pressure.
- Ref
- HYD-30
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- dy/dx
Where the depth changes slowly along a prismatic channel, the energy equation gives dy/dx = (S₀ − Sf)/(1 − Fr²), with x downstream, S₀ the bed fall, Sf the friction slope and Fr the local Froude number. The sign tells whether the surface rises or falls, and the denominator changes sign at critical flow. At Fr = 1 the formula is not defined and the flow must be treated as a control.
Rate of change of depth along a prismatic channel with the bed slope S₀, friction slope Sf and local Froude number Fr, with x increasing downstream and hydrostatic pressure. It is not defined at critical flow.
Calculator
- Numerator, S₀ − Sf4×10−4
- Denominator, 1 − Fr²0.75
- Change in depth over 100 m at this gradient0.05333 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.
- The depth rises by 0.05333 m per 100 m along the channel, since the bed fall exceeds the friction slope.
- Close to Fr = 1 the denominator goes to zero, so the gradient becomes very steep and the profile is no longer gradually varied.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| S0 Bed slope | 1e-5 to 0.1 | −7.867e-4 | 0.1325 | 0.1333 | +10% | +25 % |
| Sf Friction slope | 1e-5 to 0.1 | −0.132 | 0.00132 | 0.1333 | +10% | −15 % |
| Fr Local Froude number | 0.05 to 3 | −0.005902 | 0.00318 | 0.009082 | +10% | +7.53 % |
Sweep of Local Froude number
| Local Froude number | Depth gradient |
|---|---|
| 0.05 | 4.01e-4 |
| 0.4188 | 4.851e-4 |
| 0.7875 | 0.001053 |
| 1.156 | −0.001187 |
| 1.525 | −3.017e-4 |
| 1.894 | −1.547e-4 |
| 2.263 | −9.711e-5 |
| 2.631 | −6.753e-5 |
| 3 | −5e-5 |
Sweep one variable
Variables and how each one contributes
Depth gradient
What the equation gives you. Everything else feeds it.
Bed slope
The bed slope, positive for a fall downstream. A steeper bed raises the gradient in direct proportion to the numerator. Raising it by 10 % (to 0.0011) moves the result by +25 %, from 5.333×10−4 to 6.667×10−4.
Friction slope
The friction slope, a plain ratio. More friction lowers the numerator, so the gradient falls. Raising it by 10 % (to 6.6×10−4) moves the result by −15 %, from 5.333×10−4 to 4.533×10−4.
Local Froude number
The local Froude number (not equal to 1). A Froude number nearer 1 makes the denominator smaller, so the gradient grows rapidly. Raising it by 10 % (to 0.55) moves the result by +7.53 %, from 5.333×10−4 to 5.735×10−4.
Worked example with the values above
- Slope difference: S₀ − Sf = 0.001 − 6×10−4 = 4×10−4.
- Denominator: 1 − Fr² = 1 − 0.25 = 0.75.
- dy/dx = 5.3333×10−4 (metres per metre).
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