Gradually Varied Flow Calculator
Water-surface profile along a prismatic rectangular or trapezoidal channel by the direct step method: normal depth, critical depth, profile classification, and the distance between a control depth and any other depth — with the full step table.
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Defaults: 8-ft pool behind a structure on a mild concrete channel, computed upstream to 1.05·yₙ. Keep y₁ and y₂ in the same zone (both above yₙ, both between yₙ and yc, and so on) — a gradually varied profile cannot cross normal or critical depth.
$$ \Delta x = \frac{E_2 - E_1}{S_0 - \tfrac{1}{2}(S_{f1} + S_{f2})} \qquad E = y + \frac{Q^2}{2 g A^2} \qquad S_f = \left(\frac{n\,Q}{k\,A\,R^{2/3}}\right)^2 $$
$$ Q = \frac{k}{n} A R^{2/3} S_0^{1/2} \;\;(\text{normal depth}) \qquad \frac{Q^2 T}{g A^3} = 1 \;\;(\text{critical depth}) $$
A = (b + zy)·y flow area · P = b + 2y√(1+z²) wetted perimeter · R = A/P · T = b + 2zy top width · k = 1.486 (US) or 1.0 (SI) · g = 32.174 ft/s² or 9.80665 m/s².
Profile classification
| Profile | Depth range | Typical cause | Compute from |
|---|---|---|---|
| M1 | y > yₙ > yc | Backwater behind a dam, weir or culvert | Downstream control, upstream |
| M2 | yₙ > y > yc | Drawdown to a free overfall or steeper reach | Downstream control, upstream |
| M3 | yₙ > yc > y | Flow under a sluice gate on a mild slope | Upstream control, downstream |
| S1 | y > yc > yₙ | Pool behind an obstruction on a steep slope | Downstream control, upstream |
| S2 | yc > y > yₙ | Entry from a mild to a steep reach | Upstream control, downstream |
| S3 | yc > yₙ > y | Gate discharge onto a steep slope | Upstream control, downstream |
Worked examples
Example 1 — Backwater upstream of a check structure
Given: Rectangular concrete channel, b = 10 ft, n = 0.015, S₀ = 0.001, Q = 200 cfs. Structure holds depth at 8.0 ft.
Find: Distance upstream to where depth is within 5% of normal depth.
Normal depth: solve 200 = (1.486/0.015)·A·R2/3·√0.001 → yₙ = 3.816 ft
Critical depth: 200² × 10 / (32.174 × (10y)³) = 1 → yc = 2.317 ft
y = 8.0 > yₙ > yc → M1; end depth 1.05 × 3.816 = 4.007 ft
First of 10 steps, 8.000 → 7.601 ft: Δx = (E₂ − E₁)/(S₀ − S̄f) = -459 ft
Total with 40 steps: 6,686 ft upstream (10 steps gives 6,639 ft, 200 steps 6,691 ft — near normal depth, step count matters)
Example 2 — Trapezoidal earth channel
Given: b = 20 ft, z = 2, n = 0.025, S₀ = 0.0005, Q = 500 cfs.
yₙ = 5.21 ft · yc = 2.46 ft → mild slope; any pool above 5.21 ft produces an M1 profile
References: Chow, V.T. (1959). Open-Channel Hydraulics, McGraw-Hill, ch. 9–10. Henderson, F.M. (1966). Open Channel Flow. Chaudhry, M.H. (2008). Open-Channel Flow, 2nd ed.
Related tools
- Manning's equation — uniform flow and normal depth
- Manning's n reference card
- Culvert hydraulics — headwater and tailwater controls
- Broad-crested weir
- Stilling basin — where supercritical profiles end