Open Channel Geometry — Reference
Section properties for Manning's and critical-flow analysis. A = flow area, P = wetted perimeter, R = A/P = hydraulic radius, T = top width, D = A/T = hydraulic depth. Side slope z is horizontal:vertical (z H : 1 V), depth y, bottom width b.
Section Property Formulas
| Shape | Area A | Wetted perim. P | Hyd. radius R | Top width T |
|---|---|---|---|---|
| Rectangular | b·y | b + 2y | by/(b+2y) | b |
| Trapezoidal | (b + zy)y | b + 2y√(1+z²) | A/P | b + 2zy |
| Triangular | z·y² | 2y√(1+z²) | zy / 2√(1+z²) | 2zy |
| Circular* | (D&sub0;²/8)(θ − sinθ) | D&sub0;θ/2 | (D&sub0;/4)(1 − sinθ/θ) | D&sub0; sin(θ/2) |
*Circular (partly full), diameter D&sub0;, flow depth y: θ = 2·arccos(1 − 2y/D&sub0;) radians. Full pipe: A = πD&sub0;²/4, R = D&sub0;/4.
Manning's Equation
V = Q/A = (k/n) R2/3 S1/2. S is the channel (friction) slope; n is Manning's roughness. Normal depth is found by solving Manning's for the y that passes the design Q — iterative for all shapes except the simplest.
Best Hydraulic Section (Max Q for Given Area)
| Shape | Optimum condition | R at optimum |
|---|---|---|
| Rectangular | b = 2y (width = twice depth) | y/2 |
| Trapezoidal | Half-hexagon: z = 1/√3 (60° sides) | y/2 |
| Triangular | z = 1 (90° vee, sides at 45°) | y/(2√2) |
| Semicircle | Overall most efficient open shape | y/2 |
Sources: Chow, V.T. (1959), Open-Channel Hydraulics, Table 2-1. Sturm, T.W., Open Channel Hydraulics. Standard prismatic-channel geometry.
Worked example: trapezoidal grass channel
b = 6 ft, side slopes 3H:1V, depth 2.0 ft, n = 0.035, S = 0.2%
For a vegetated channel, the next checks are permissible velocity and shear stress. Velocity of 2.25 ft/s is well within what established grass tolerates. Bare soil during establishment is another matter, which is why the temporary lining often governs the design. Also check freeboard at the design flow, and check capacity with the lower n of mowed grass and the higher n of tall grass. The two give different answers for depth and for velocity.
Common mistakes
- Writing side slope as V:H. On this card and in most civil practice, z is horizontal per one vertical. A 3:1 slope is z = 3, not z = 1/3.
- Using the triangular formula for a trapezoid with a narrow bottom. A 2-ft bottom is still a trapezoid, and dropping b from the area understates capacity noticeably at shallow depth.
- Computing the circular section with θ in degrees. The formulas need radians.
- Assuming a circular conduit flows at its maximum capacity full. Maximum capacity is near 93% depth and maximum velocity near 81% depth. See the storm sewer card for why to design to full flow anyway.
When this card doesn’t apply
These formulas describe prismatic channels with one roughness. A natural stream with a main channel and floodplains needs to be split into subsections, or the composite R badly understates conveyance. Channels lined with different materials on the bed and the banks need a composite n. Bends, transitions and non-uniform flow need a water-surface profile, not a single normal-depth calculation.
Related cheat sheets and tools
Feed these section properties into the Manning's tool with an n value, check the flow regime with the Froude / hydraulic-jump card, and protect the channel with riprap sizing. For a backwater profile along one reach, use the gradually varied flow (direct step) calculator; the riprap sizing calculator runs the USACE and Isbash methods. For modeling a channel network within a full watershed, see HydroComplete.