Open Channel Flow (Manning-Strickler)

Flow velocity · Discharge Q · Bed slope · Hydraulic radius · Froude number

Open Channel Calculator

Select calculation:

Flow velocity v (R direct input)

Rectangular channel – Q, v, Fr

Trapezoidal channel – Q, v, Fr

Circular pipe (partial flow) – Q, v, Fr

Back-calculate bed slope I

Strickler coefficient k_st [m^1/3/s]

Bed slope I (dimensionless)

= 1.0 ‰

Example: 0.001 = 1 ‰ | 0.005 = 5 ‰

Hydraulic radius R [m] R = A / P (cross-sectional area / wetted perimeter)

Formulas & Fundamentals

Manning-Strickler – flow formula:
v = k_st · R^2/3 · I^1/2 [m/s]
k_st = Strickler coefficient [m^1/3/s]
R = hydraulic radius [m] | I = bed slope [–]

Discharge & hydraulic radius:
Q = v · A [m³/s]
R = A / P [m]
A = cross-sectional area | P = wetted perimeter

Froude number (flow regime):
Fr = v / √(g · D_h)
D_h = A/T (hydraulic depth) | T = top water width
Fr < 1: subcritical flow (tranquil)
Fr > 1: supercritical flow (rapid)

Cross-section formulas

Rectangle:
A = b·y | P = b + 2y | T = b

Trapezoid (side slope m = horiz./vert.):
A = (b + m·y)·y
P = b + 2y·√(1 + m²) | T = b + 2m·y

Circle (partial flow, θ in rad):
θ = 2·arccos(1 – 2h/D)
A = D²/8·(θ – sin θ)
P = D·θ/2 | T = D·sin(θ/2)

Strickler coefficient k_st (typical values)

Material / Condition	k_st
Plastic, glass	90–100
Concrete smooth / precast	80–90
Concrete normal	70–80
Masonry rendered	55–65
Earth channel, well maintained	40–50
Natural channel, clean	28–40
Natural channel, vegetated	15–28

Conversion: k_st = 1/n (n = Manning roughness)

Open Channel Flow – Fundamentals of Free-Surface Hydraulics

What does the Manning-Strickler formula describe?

The Manning-Strickler formula is the most widely used empirical equation for uniform, steady flow in open channels (canals, ditches, rivers, partially filled pipes). It relates the mean flow velocity v to the bed slope I, the hydraulic radius R, and the roughness coefficient k_st:

v = k_st · R^2/3 · I^1/2
Discharge follows as Q = v · A. The bed slope I = Δh/L describes the elevation difference per unit length of channel.

Flow regimes – Froude number Fr

Fr	Flow regime
Fr < 1	subcritical (tranquil, backwater possible)
Fr = 1	critical flow
Fr > 1	supercritical (rapid, shooting)

Strickler vs. Manning

European standard: Strickler coefficient k_st [m^1/3/s]
Anglo-Saxon: Manning roughness n [s/m^1/3]

Conversion: k_st = 1/n
Example: n = 0.013 → k_st ≈ 77

Cross-section shapes and their hydraulic properties

Rectangular channel

Simple calculation: A = b·y, P = b + 2y.
Hydraulically optimal at b/y = 2 (half-square): R = y/2 is then maximum.

Trapezoidal channel

Earth canals: m ≥ 1.0–1.5 (stability).
Hydraulically optimal: m = 1/√3 ≈ 0.577 (equilateral trapezoid).
Half-circle is the ideal hydraulic section.

Circular pipe (partial)

Max. v at h/D ≈ 81 %.
Max. Q at h/D ≈ 94 %.
(More favourable R/A ratio than full-bore flow.)

Worked example: trapezoidal earth channel

Problem:

Earth canal: bottom width b = 1.5 m, water depth y = 1.0 m, side slope m = 1.5, k_st = 35 m^1/3/s, bed slope I = 0.0005.

Solution:

A = (1.5 + 1.5·1.0)·1.0 = 3.00 m²
P = 1.5 + 2·1.0·√(1 + 1.5²) = 1.5 + 3.606 = 5.106 m
R = 3.00 / 5.106 = 0.5876 m
v = 35 · 0.5876^2/3 · √0.0005 = 35 · 0.6805 · 0.02236 = 0.532 m/s
Q = 0.532 · 3.00 = 1.60 m³/s
T = 1.5 + 2·1.5·1.0 = 4.5 m → D_h = 3.00/4.5 = 0.667 m
Fr = 0.532 / √(9.81·0.667) = 0.532 / 2.559 = 0.208 → subcritical flow

Frequently Asked Questions

The hydraulic radius R = A/P is the ratio of the wetted cross-sectional area A to the wetted perimeter P. Only for a semicircular section does R = y/2 hold. For a very wide, shallow channel (b ≫ y), R ≈ y approximately. The larger R, the less wall friction per unit cross-section – a large R means a hydraulically efficient section.

At h/D ≈ 94 % fill, discharge Q reaches its maximum because the hydraulic radius R at partial fill can be more favourable than for the full cross-section (R_full = D/4). Maximum flow velocity already occurs at h/D ≈ 81 %. This effect is technically important: overloaded sewers lose capacity when transitioning from free-surface to pressurised flow.

k_st is determined by the material and condition of the channel bed and walls. For capacity checks (verifying sufficient discharge capacity), a lower k_st governs (unfavourable – less capacity). For erosion checks, a higher k_st gives higher velocity, which is unfavourable. Normative reference values: EN 752, ATV-DVWK A 110, DVWK Guidelines 220.

The Froude number Fr compares the flow velocity to the speed of shallow water waves (c = √(g·D_h)). Subcritical flow (Fr < 1): tranquil, the water surface can respond to backwater conditions (typical: rivers, drains). Supercritical flow (Fr > 1): rapid, information travels only downstream (typical: stilling basins, steep chutes). The transition occurs via a hydraulic jump.

The Strickler formula strictly applies only to uniform, steady flow (normal depth: energy slope = bed slope = water-surface slope). For backwater and water-depth changes along the channel it is applied reach-by-reach (backwater curve computation, gradually varied flow equation). As an approximation it is well suited to mildly non-uniform flow when the local energy slope is used.

Summary

Strickler formula

v = k_st·R^2/3·I^1/2
Q = v · A

Hydraulic radius

R = A / P
Measure of hydraulic efficiency

Froude number

Fr = v / √(g·D_h)
Fr < 1: subcritical

Typical applications

Urban drainage: sizing of stormwater and wastewater sewers
Hydraulic engineering: design of ditches, culverts, flood channels
Flood protection: discharge capacity of watercourses and retention areas
Agriculture: field drainage ditches, irrigation canals, land improvement
Dam engineering: stilling basins, spillway chutes, overflow weirs

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