Open-channel hydraulics

Why it matters

Once you know a peak flow, you must carry it somewhere: in a roadside ditch, a storm sewer, a culvert or a stream channel. Most of these flow with a free surface open to the air (a pipe flowing partly full counts too). Open-channel hydraulics answers how deep the water will be, how fast it moves, and whether the channel will erode or overflow. Manning's equation is the everyday tool.

Geometry first

For a channel cross-section with flow depth y:

• Flow area A.
• Wetted perimeter P: the length of boundary in contact with water (not the free surface).
• Hydraulic radius R = A / P.
• Top width T at the water surface.

For a trapezoid with bottom width b and side slopes z horizontal to 1 vertical:

A = (b + z*y) * y
P = b + 2*y*sqrt(1 + z^2)
T = b + 2*z*y

A rectangle is the special case z = 0. Circular pipes have their own (longer) formulas, and design charts or software handle partly full pipes.

Manning's equation

For uniform flow, where depth and velocity do not change along the channel, gravity's pull down the slope balances boundary friction:

V = (k / n) * R^(2/3) * S^(1/2)        Q = V * A
k = 1.0 (SI: m, m/s)      k = 1.49 (US: ft, ft/s)

• S is the channel bed slope (m/m or ft/ft); in uniform flow it equals the energy slope.
• n is Manning's roughness coefficient, an empirical value for the lining: roughly 0.011 to 0.015 for smooth concrete and pipe, 0.02 to 0.03 for earth channels, and higher for weedy or rocky streams. Published tables (for example in FHWA and USGS references, or your agency's manual) give ranges; choosing n is a judgement call that matters as much as the arithmetic.

Worked example (illustrative)

A concrete-lined trapezoidal channel has b = 2.0 m, side slopes z = 2 (2H:1V), n = 0.015 and S = 0.001. At a depth of 1.0 m:

1. A = (2.0 + 2 × 1.0) × 1.0 = 4.00 m^2
2. P = 2.0 + 2 × 1.0 × sqrt(5) = 6.47 m
3. R = 4.00 / 6.47 = 0.618 m
4. V = (1 / 0.015) × 0.618^(2/3) × 0.001^(1/2) = 66.7 × 0.726 × 0.0316 = 1.53 m/s
5. Q = 1.53 × 4.00 = 6.12 m^3/s

Normal depth

Usually you know Q and need the depth. Manning's equation cannot be rearranged for y in closed form for most shapes, so you solve it numerically: guess a depth, compute Q, and adjust. Because Q increases steadily with depth, bisection always works: start with a depth that is too small and one that is too large, test the midpoint, and keep the half that brackets the answer. Spreadsheets use Goal Seek; Python can use bisection or scipy.optimize.brentq in a full environment. The resulting depth is the normal depth yn. Add freeboard above it so waves, debris and uncertainty in n do not cause overtopping.

Subcritical or supercritical: the Froude number

Fr = V / sqrt(g * D)      D = A / T (hydraulic depth)

• Fr < 1: subcritical (tranquil). Most natural streams and mild channels. Downstream conditions affect the depth.
• Fr > 1: supercritical (rapid). Steep chutes and spillways. Waves cannot travel upstream.
• Fr = 1: critical flow; the depth is unstable and surface waves form.

For the example, D = 4.00 / 6.0 = 0.667 m and Fr = 1.53 / sqrt(9.81 × 0.667) = 0.60: subcritical. A change from supercritical to subcritical flow forms a hydraulic jump, which dissipates energy and must be controlled with an energy dissipator.

Design checks

• Capacity: normal depth plus freeboard fits within the section.
• Velocity limits: high velocities erode unlined or grass channels; very low velocities let sediment settle in pipes (minimum self-cleansing velocities, often around 0.6 to 0.9 m/s, appear in many manuals).
• Lining shear stress: tau = gamma * R * S compared with the permissible shear of the lining (the method used in FHWA's HEC-15).
• Culverts and bridges have their own inlet and outlet control methods (FHWA HDS-5, HEC-RAS).

Common mistakes

• Using k = 1.0 with US units, or 1.49 with SI.
• Including the free surface in the wetted perimeter.
• Using percent slope (0.1) instead of decimal (0.001).
• Treating n as exact: test a range, especially for natural channels.
• Applying uniform-flow results near a culvert, drop or junction where flow is gradually or rapidly varied.

Professional responsibility

Channel, culvert and storm-sewer designs follow the reviewing agency's manual and are sealed by licensed engineers. These values are illustrative. Real designs need review and sealing by a licensed professional engineer (PE) under the codes your jurisdiction adopts.

Recap

• R = A / P; Manning: V = (k/n) R^(2/3) S^(1/2), Q = V A.
• Solve normal depth numerically; bisection is simple and reliable.
• Check Froude number, velocity, shear and freeboard before calling a channel adequate.

Try it

Open the Do tab to code trapezoid geometry, Manning's equation, a normal-depth solver and the Froude number.