roadside ditch trapezoidal grass-lined North Carolina

Sizing a trapezoidal roadside channel: Manning's normal depth, critical depth, and Froude number

A roadside ditch along a new commercial driveway has to carry the 10-year flow without overtopping or scouring out. This solves it the way a reviewer can check by hand: Manning's equation for normal depth (iterated), the resulting velocity, the critical depth and Froude number to fix the flow regime, and a permissible-velocity check on the grass lining. No proprietary charts — every number reproduces from the equations.

Result: For 25 cfs in a 2-ft-bottom, 3:1-side-slope channel at 1.0% on n = 0.030, the normal depth is 1.17 ft at a velocity of 3.9 ft/s. Critical depth is 1.05 ft and the Froude number is 0.81, so flow is subcritical (y_n > y_c). 3.9 ft/s is acceptable for an established grass lining but marginal for bare earth — add a liner or flatten the grade if the channel won't be vegetated at startup. Step-by-step below.

Channel inputs

The geometry and hydraulics an engineer actually fills out:

Parameter	Value	Source
Design discharge, Q	25 cfs	10-yr peak from watershed analysis
Bottom width, b	2.0 ft	Channel cross-section
Side slope, z (H:V)	3:1	Mowable, stable side slope
Longitudinal slope, S₀	0.010 ft/ft (1.0%)	Profile grade
Manning's n	0.030	Established grass lining, good condition

Step 1 — Manning's equation and channel geometry

Manning's equation in US customary units:

$Q = \frac{1.49}{n}\, A\, R^{2/3}\, S_0^{1/2}$

For a trapezoid with bottom width b and side slope z (horizontal:vertical), at depth y:

$A = by + zy^2 = 2y + 3y^2$

$P = b + 2y\sqrt{1+z^2} = 2 + 2y\sqrt{10} = 2 + 6.325y$

$R = \frac{A}{P}$

Collecting the constants, (1.49/n)·S₀^½ = (1.49/0.030)(0.010)^½ = 49.67 × 0.1 = 4.967, so normal depth is the y that satisfies:

$A\,R^{2/3} = \frac{Q}{4.967} = \frac{25}{4.967} = 5.033$

Step 2 — Normal depth by iteration

There's no closed form for y in a trapezoid, so iterate on A·R^(2/3) until it hits 5.033:

Trial y (ft)	A (ft²)	P (ft)	R (ft)	A·R^2/3
1.15	6.268	9.273	0.676	4.83
1.20	6.720	9.590	0.701	5.30
1.17	6.447	9.400	0.686	5.01

At y = 1.17 ft, A·R^2/3 = 5.01 ≈ 5.033. Normal depth y_n = 1.17 ft.

Step 3 — Velocity and capacity

With A = 6.447 ft² at normal depth:

$V = \frac{Q}{A} = \frac{25}{6.447} = 3.88 \text{ ft/s}$

Top width and the freeboard you'd carry to the channel lip:

$T = b + 2zy = 2 + 6(1.17) = 9.02 \text{ ft}$

NCDOT roadside channels typically carry 0.5 ft of freeboard, so build the section to ~1.67 ft total depth (≈ a 2-ft-deep ditch).

Step 4 — Critical depth and flow regime

Flow is critical when the Froude number equals 1, i.e., when Q²·T = g·A³. Solve for the critical depth y_c:

$\frac{Q^2}{g} = \frac{A^3}{T} \;\Rightarrow\; \frac{25^2}{32.2} = 19.41 = \frac{A^3}{T}$

Iterating the trapezoid geometry, y_c ≈ 1.05 ft gives A = 5.41 ft², T = 8.30 ft, A³/T = 19.1 ≈ 19.41. Critical depth y_c = 1.05 ft.

Froude number at normal depth, with hydraulic depth D = A/T = 6.447/9.02 = 0.715 ft:

$Fr = \frac{V}{\sqrt{gD}} = \frac{3.88}{\sqrt{32.2 \times 0.715}} = \frac{3.88}{4.80} = 0.81$

Fr = 0.81 < 1, and y_n (1.17) > y_c (1.05): the flow is subcritical — both checks agree, as they must. Subcritical flow is controlled from downstream, so the tailwater at the outlet sets the profile.

Reading check: normal depth above critical depth always means subcritical, and Fr must come out below 1. If your Fr lands above 1 while y_n > y_c, you've mixed up A/T (hydraulic depth) with the max depth — Froude uses A/T, not y.

Step 5 — Lining and erosion check

The computed 3.9 ft/s has to be below the permissible velocity for the lining:

Lining	Permissible velocity	Verdict at 3.9 ft/s
Bare earth (silt loam)	~2.5 – 3.5 ft/s	Exceeded — will scour
Established grass, good cover	~4 – 5 ft/s	OK
Turf reinforcement mat / riprap	> 6 ft/s	OK with margin

The design is fine once the grass is established, but bare at construction startup it would erode. Specify a temporary liner (erosion-control blanket) until vegetation takes, or flatten the grade. Dropping S₀ to 0.5% drops the velocity to ~3.1 ft/s and raises normal depth — see the table below.

What changes if you tweak the inputs

If you change…	The result moves…
Slope 1.0% → 0.5%	y_n rises to ~1.34 ft; V drops to ~3.1 ft/s (gentler, deeper, slower)
n 0.030 → 0.040 (heavier grass)	y_n rises to ~1.32 ft; V drops to ~3.2 ft/s
Bottom width 2 ft → 4 ft	y_n drops to ~0.95 ft; wider, shallower section
Q 25 → 40 cfs	y_n rises to ~1.49 ft; V to ~4.3 ft/s — re-check the lining
Side slope 3:1 → 2:1	Slightly deeper normal depth, narrower top width

Open this channel in HydroComplete

The Conveyance engine solves Manning's normal depth, critical depth, and Froude number for 14 channel/closed-conduit shapes. Change the slope, the lining n, or the geometry and watch depth, velocity, and regime update instantly.

Open in app 14-day free trial

Sources and further reading

Chow, V. T. (1959). Open-Channel Hydraulics. McGraw-Hill — Manning's equation, normal and critical depth.
USDA-NRCS. National Engineering Handbook, Part 654 — Stream Restoration Design. Manning's n values.
FHWA. Hydraulic Design of Highway Culverts (HDS-5). Open-channel approach and tailwater.
NCDOT. Guidelines for Drainage Studies and Hydraulic Design. Roadside channel and permissible-velocity criteria.

— Michael Flynn, PE
This worked example uses HydroComplete's Conveyance engine for the Manning, normal-depth, and critical-depth solvers. Open the scenario in the app to verify or modify any input.

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