Topic 6 of 10 — free theory
Open-Channel Flow
Manning's equation, normal depth, critical depth, and the Froude number.
Take the free 5-question mini-quiz ↓
Manning's equation — the workhorse of open channels
For uniform flow in an open channel (rivers, canals, sewers flowing partly full), Manning's equation relates discharge to the channel's shape, roughness, and slope:
V = (k/n) · R2/3 · S1/2 Q = AV
| V | mean velocity |
| Q | discharge |
| k | unit constant: 1.486 for English units (ft/s), 1.0 for SI |
| n | Manning's roughness coefficient (concrete ≈ 0.013, earth ≈ 0.022, natural channel ≈ 0.03–0.05) |
| R | hydraulic radius = A/P, where P is the wetted perimeter |
| S | longitudinal slope of the channel (energy slope for uniform flow) |
Uniform flow means depth and velocity are constant along the channel — the water surface parallels the bed. Normal depth is simply the uniform-flow depth for a given Q.
Geometry you will need on the fly
Rectangular: A = by, P = b + 2y Trapezoidal: A = (b + zy)y, P = b + 2y√(1+z²) Full circular: R = D/4
| b | bottom width |
| y | flow depth |
| z | side slope, horizontal:vertical |
| Do NOT use R = D/4 for a partly full pipe — recompute A and P for the actual depth. |
Critical flow and the Froude number
Open-channel flow has two regimes. Subcritical (slow, deep — Fr < 1) is controlled from downstream; supercritical (fast, shallow — Fr > 1) is controlled from upstream. The boundary is critical flow:
Fr = V / √(g·Dh) with Dh = A/T
| Fr | Froude number: < 1 subcritical, = 1 critical, > 1 supercritical |
| Dh | hydraulic depth = area / top width T (for a rectangle, Dh = y) |
Rectangular channels: yc = (q²/g)1/3 and Emin = 3yc/2
| yc | critical depth |
| q = Q/b | discharge per unit width |
| E = y + V²/2g | specific energy — minimised at critical depth |
Specific energy E is measured from the channel bed. At a given E, two alternate depths exist (one subcritical, one supercritical) — the exam loves asking which is which.
PE depth: hydraulic jumps, culverts, and controls
A hydraulic jump converts supercritical flow to subcritical flow and dissipates energy. For a rectangular channel, conservation of momentum gives the sequent-depth relationship:
y2y1 = 0.5[√(1 + 8Fr1²) − 1] ΔE = (y2−y1)³4y1y2
For culverts, first identify the control. Inlet control depends mainly on entrance geometry and headwater. Outlet control requires the full energy balance, including barrel friction, entrance/exit losses, tailwater, and elevation. The controlling case is the one requiring the higher headwater for the design flow.
Stormwater link: gutter, inlet, and storm-sewer design couples surface capture with closed-conduit capacity. Bypass flow from one inlet becomes approach flow to the next; never size each inlet as though it receives only local runoff.
Worked example Normal depth in a trapezoidal channel
Given:
- Trapezoidal channel: bottom width b = 6 ft, side slopes 2H:1V (z = 2).
- Manning's n = 0.013, bed slope S = 0.001, discharge Q = 60 cfs.
Solution:
- Normal depth needs trial: guess y, compute A, P, R, then Q = (1.486/n)·A·R2/3·S1/2.
- y = 1.50 ft → A = 13.50 ft², P = 12.71 ft, R = 1.062 ft → Q = 50.8 cfs (low).
- y = 1.60 ft → A = 14.72 ft², P = 13.16 ft, R = 1.119 ft → Q = 57.3 cfs (low).
- y = 1.65 ft → A = 15.35 ft², P = 13.38 ft, R = 1.147 ft → Q = 60.8 cfs (just high).
- Interpolating: yn ≈ 1.64 ft. On the exam, bracket the answer between two trials — you rarely need more than two or three guesses.
Answer: Normal depth yn ≈ 1.64 ft.