Channel Analysis

Open Channel Analysis

Analyse uniform flow in open channels and partially-full pipes using Manning’s equation. This guide covers the five supported cross-section geometries, the four solve modes, Froude-number flow regimes, critical depth, boundary shear stress, freeboard, and a step-by-step worked example for sizing a trapezoidal drain.

Overview

The Channel Analysis tool calculates uniform (normal) flow conditions in open channels and partially-full pipes using the Manning equation. It supports five channel geometries — trapezoidal, rectangular, circular, triangular, and arbitrary station-elevation cross-sections — and four solve modes that let you enter the depth, the flow, a maximum velocity, or a maximum shear stress and back-calculate the rest.

Results include the full set of hydraulic variables needed for design: area of flow, wetted perimeter, hydraulic radius, top width, hydraulic depth, velocity, Froude number, critical depth and slope, average and maximum boundary shear stress, and a recommended freeboard. All calculations run instantly in the browser — no server round-trip or credits required — and results can be saved and linked to existing projects.

Uniform flow assumption

The tool assumes uniform flow: the water surface is parallel to the bed, energy grade line, and channel bottom, so the friction slope equals the bed slope. Real channels only approach uniform flow on long, prismatic reaches far from transitions, controls, or changes in geometry. Where the water surface is not parallel to the bed (backwater, drawdown, hydraulic jump), use the GVF Profile tool instead.

Channel Types

The tool supports five cross-section geometries. Select the one that matches the channel you are analysing; the formulas below show how area of flow A, wetted perimeter P, and top width T are computed as functions of the depth y.

Trapezoidal

The most common prismatic open-channel shape: flat bed of width B with two planar side slopes expressed as horizontal-to-vertical ratios Z₁ and Z₂. Asymmetric side slopes are fully supported — set Z₁ ≠ Z₂ to model one-sided cut/fill situations or natural channels with unequal banks.

$$A = B\,y + \tfrac{Z_1 + Z_2}{2}\,y^2 \qquad P = B + y\sqrt{1 + Z_1^{2}} + y\sqrt{1 + Z_2^{2}} \qquad T = B + (Z_1 + Z_2)\,y$$

Trapezoidal geometry

Rectangular

A special case of the trapezoidal shape with vertical walls (Z₁ = Z₂ = 0). Defined by bottom width B alone. Used for lined concrete or masonry channels, box culverts flowing with a free surface, and laboratory flumes.

$$A = B\,y \qquad P = B + 2y \qquad T = B$$

Rectangular geometry

Circular

For partially-full pipe flow. Defined by the internal pipe diameter D. The water-surface subtended angle θ (in radians) is computed from the depth-to-diameter ratio, and the area and wetted perimeter follow from circular-segment geometry.

$$\theta = 2\arccos\!\left(1 - \tfrac{2y}{D}\right) \qquad A = \tfrac{D^{2}}{8}\,(\theta - \sin\theta) \qquad P = \tfrac{D\,\theta}{2} \qquad T = D\sin(\theta/2)$$

Circular pipe geometry (partially full)

Circular pipe capacity

Peak discharge in a circular pipe occurs at roughly y/D ≈ 0.938, not at full flow (y = D). Above this depth the wetted perimeter grows faster than the area, so A·R^(2/3) — and therefore Q — decreases. When you enter a flow and solve for depth, the tool searches for solutions below the peak so you get the normal operating depth rather than the unstable branch.

Triangular

V-shaped channel with no bottom width — common for roadside ditches, small median drains, and V-notch flumes. Defined by side slopes Z₁ and Z₂. Asymmetric slopes are supported.

$$A = \tfrac{Z_1 + Z_2}{2}\,y^2 \qquad P = y\sqrt{1 + Z_1^{2}} + y\sqrt{1 + Z_2^{2}} \qquad T = (Z_1 + Z_2)\,y$$

Triangular geometry

Custom cross-section

For natural or irregularly shaped channels, define an arbitrary cross-section using station-elevation pairs. Paste data from a survey, HEC-RAS export, or GIS cross-section extraction (tab-, comma-, or space-separated, one point per line). The tool integrates area and wetted perimeter over the submerged portion of the profile at the current water-surface elevation.

Composite roughness

Manning’s n can vary across a natural section (main channel vs. vegetated floodplain). The current version of the tool uses a single n applied to the entire wetted perimeter. If your section has strongly varying roughness, either analyse sub-sections separately or use a composite-n formulation (e.g. Horton, Einstein) and enter the weighted value manually.

Manning’s Equation

Manning’s equation relates discharge to channel geometry, roughness, and longitudinal slope for uniform flow. It is the workhorse equation of open-channel hydraulics and applies equally to lined channels, natural streams, and partially-full conduits.

$$Q \;=\; \tfrac{1}{n}\,A\,R^{2/3}\,S^{1/2}$$

Manning's equation — SI (metric) units

$$Q \;=\; \tfrac{1.486}{n}\,A\,R^{2/3}\,S^{1/2}$$

Manning's equation — US customary units

Where:

• Q — discharge (m³/s in SI; ft³/s in US customary)
• n — Manning’s roughness coefficient (dimensionless — strictly, it has units, but they are conventionally absorbed)
• A — cross-sectional area of flow (m² or ft²)
• R — hydraulic radius = A/P (m or ft)
• S — longitudinal bed slope (m/m or ft/ft) — equal to the friction slope under uniform flow
• P — wetted perimeter (m or ft)

The factor 1.486 in the US form is the conversion (3.281)^(1/3), which appears because the hydraulic radius is raised to the 2/3 power when converting between metres and feet. The tool works internally in SI; US-customary inputs are converted on entry and converted back on display.

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

Hydraulic radius

Continuity relates discharge to mean velocity:

$$Q \;=\; V \cdot A \qquad \Longleftrightarrow \qquad V \;=\; \frac{Q}{A}$$

Continuity

Manning’s n values

The tool includes a built-in reference with over 50 common channel and pipe materials organised into four categories. Click the Lookup button next to the Manning’s n input to browse and select a value — you can click on the minimum, typical, or maximum column to populate the input.

Category	Surface	n (min)	n (typ)	n (max)
Pipes	Smooth concrete / PVC / HDPE	0.010	0.012	0.014
Pipes	Corrugated metal pipe (CMP)	0.020	0.024	0.027
Pipes	Vitrified clay / brick	0.011	0.013	0.017
Lined channels	Smooth concrete (trowelled)	0.011	0.013	0.015
Lined channels	Float-finished / unfinished concrete	0.013	0.015	0.017
Lined channels	Shotcrete (gunite)	0.016	0.019	0.023
Lined channels	Dry-stone / rubble masonry	0.017	0.025	0.030
Lined channels	Asphalt	0.013	0.016	0.018
Excavated channels	Earth, straight, uniform, clean	0.016	0.022	0.030
Excavated channels	Earth, winding, some weeds / stones	0.023	0.030	0.040
Excavated channels	Earth, dense weeds or deep pools	0.030	0.040	0.050
Excavated channels	Gravel bed, clean	0.022	0.025	0.030
Excavated channels	Cobble / boulder bed	0.030	0.040	0.050
Natural channels	Small clean straight stream (plain)	0.025	0.030	0.033
Natural channels	Small stream, weeds / pools	0.033	0.040	0.045
Natural channels	Mountain stream, cobbles / boulders	0.040	0.050	0.070
Natural channels	Floodplain — short grass / pasture	0.025	0.030	0.035
Natural channels	Floodplain — dense brush / timber	0.080	0.100	0.160

Selecting Manning's n

When in doubt, use the typical value. Use minimum values for new, well-maintained channels and pipes; use maximum values for aged, poorly maintained channels with vegetation growth, debris, or silt deposits. For critical designs, run the calculation with both the upper and lower bounds of n to see the sensitivity — Q is inversely proportional to n, so a 20% error in n directly translates into a 20% error in capacity (or depth, when solving inversely).

Normal depth

Normal depth y_n is the depth at which uniform flow occurs — where the gravitational force driving the flow is exactly balanced by boundary friction. At normal depth, S_f = S_0: the friction slope equals the bed slope. When you enter a discharge and solve for depth (the most common design case), the tool finds y_n by inverting Manning’s equation:

$$\tfrac{1}{n}\,A(y_n)\,R(y_n)^{2/3}\,S^{1/2} \;-\; Q \;=\; 0$$

Normal depth — implicit

Because A(y) and R(y) are non-linear functions of depth for every geometry except the infinitely wide rectangle, there is no closed-form inverse. The tool uses a bisection method over a bracket [y_min, y_max] and iterates until the residual is below 10⁻⁶ m³/s. For circular pipes the bracket is clamped below the capacity-peak depth (y/D ≈ 0.938) so the solver returns the stable lower-depth solution.

Geometric properties

Five geometric properties of the flow section drive every other result. The tool reports all of them in the results card.

Property	Symbol	Definition
Area of flow	A	Cross-sectional area below the water surface
Wetted perimeter	P	Length of channel boundary in contact with water (excludes the free surface)
Hydraulic radius	R = A/P	Ratio of area to wetted perimeter — the length scale that appears in Manning’s equation
Top width	T	Width of the water surface
Hydraulic depth	D_h = A/T	Mean depth — used in the Froude number

The hydraulic radius R is the most important of these in flow computations: it captures both the size of the cross-section and its efficiency (how much of the perimeter is boundary vs. free surface). For a very wide rectangular channel R ≈ y; for a circular pipe flowing half full R = D/4; for a narrow-deep rectangle R < y/2.

Froude Number and Flow Regime

The Froude number Fr is the ratio of inertial to gravitational forces in the flow, and it classifies the flow regime as subcritical, critical, or supercritical. It is computed from the mean velocity and the hydraulic depth:

$$Fr \;=\; \frac{V}{\sqrt{g\,D_h}} \qquad \text{where } D_h = \frac{A}{T}$$

Froude number

Froude number	Flow regime	Characteristics
Fr < 1	Subcritical	Deep, slow flow. Gravity dominates. Downstream conditions control. Small disturbances (waves) travel both upstream and downstream. Typical of natural rivers and mild-slope channels.
Fr = 1	Critical	Transition between regimes. Specific energy is minimised for a given discharge. Unstable — small perturbations produce large water-surface fluctuations. Avoid as a design state.
Fr > 1	Supercritical	Shallow, fast flow. Inertia dominates. Upstream conditions control. Disturbances cannot propagate upstream. Typical of steep chutes and spillways. A jump to subcritical flow (hydraulic jump) usually occurs where the channel flattens.

Designing a channel at or very near Fr = 1 is undesirable because the water surface becomes unstable and the transition between regimes can produce cross-waves, surging, and increased erosion. SANRAL recommends Fr ≤ 0.85 for subcritical channels and Fr ≥ 1.15 for supercritical channels where practical.

Specific energy and alternate depths

For a given discharge in a prismatic channel, specific energy E = y + V²/(2g) is a function of depth only. It has a minimum at critical depth y_c, and the same E (above E_min) is shared by two alternate depths — one subcritical, one supercritical. Which one actually occurs is set by channel slope and boundary controls. This is why slope-matters: a channel of a given Q and n on a mild slope (S < S_c) carries the flow at a subcritical normal depth, while the same Q and n on a steep slope (S > S_c) carries it at a supercritical normal depth. The tool reports both the normal and critical states so you can see which side of critical you are on.

Critical flow

Critical depth y_c is the depth at which Fr = 1. It is independent of slope and roughness — it depends only on discharge and channel geometry — and satisfies:

$$\frac{Q^{2}\,T(y_c)}{g\,A(y_c)^{3}} \;=\; 1$$

Critical-flow condition

The tool solves this iteratively. Once y_c is known, critical velocity and critical slope follow directly:

$$V_c \;=\; \frac{Q}{A_c} \qquad S_c \;=\; \left(\frac{n\,Q}{A_c\,R_c^{2/3}}\right)^{2}$$

Critical velocity and critical slope

Compare the design bed slope S to S_c to classify the channel:

• S < S_c — mild slope: normal flow is subcritical (y_n > y_c).
• S > S_c — steep slope: normal flow is supercritical (y_n < y_c).
• S ≈ S_c — critical slope: avoid; normal flow is unstable.

Shear Stress

Boundary shear stress drives erosion. It must be checked against the permissible shear stress for the channel lining or soil, especially when sizing unlined earth channels, grass-lined swales, or designing rip-rap and mattresses.

$$\tau_\text{avg} = \gamma\,R\,S \qquad\qquad \tau_\text{max} = \gamma\,y\,S$$

Average and maximum boundary shear stress

Where γ = ρg ≈ 9810 N/m³ is the specific weight of water. The average shear stress is distributed over the entire wetted perimeter and is the correct value to compare against bank-averaged permissible stresses. The maximum bed shear stress applies to the deepest point of the section and is the conservative value for bed-material stability checks.

Typical permissible shear stresses (for reference — verify against your design standard):

Lining	τ_permissible (N/m²)
Fine sand / silt	1 – 3
Sandy loam	2 – 4
Firm clay	10 – 30
Gravel (d₅₀ = 25 mm)	~20
Cobble (d₅₀ = 100 mm)	~80
Short grass (good cover)	30 – 50
Long grass (good cover)	80 – 180
Rip-rap (d₅₀ = 150 mm)	~120
Concrete	> 1000

If τ_avg (or τ_max for a point check) exceeds the permissible value, increase the channel size, reduce the slope, or upgrade the lining. The Channel Lining Design tool automates this check for common lining types including grass, rip-rap, and turf-reinforcement mats.

Freeboard

Freeboard is the additional channel depth above the design water surface provided to accommodate waves, surges, super-elevation on bends, debris, and uncertainty in the design flow. The tool computes a recommended freeboard from flow magnitude and velocity head, following SANRAL and USBR guidance:

Design flow	Recommended minimum freeboard
Q < 0.5 m³/s	0.15 m
0.5 – 1.5 m³/s	0.30 m
1.5 – 85 m³/s	0.45 m + velocity head V²/(2g)
> 85 m³/s	0.60 m + velocity head V²/(2g)

Freeboard is a minimum

The freeboard values above are minimums. Add extra depth where the channel curves (super-elevation), where debris or sediment is expected, where lining transitions occur, or where the consequences of overtopping are severe (e.g. adjacent to a highway or urban area). For supercritical flow in lined channels, SANRAL recommends checking the standing-wave height at bends using Δy = V² · B / (g · r_c).

Solve Modes

The tool supports four solve modes. Choose the one that matches your design question.

Mode	Input	Solves for	Method
Enter Depth	Depth y (m)	Flow, velocity, and all derived quantities	Direct substitution into Manning’s equation
Enter Flow	Discharge Q (m³/s)	Normal depth y_n and derived quantities	Bisection on Q(y) - Q_target = 0
Max Velocity	V_max (m/s)	Depth and flow at which V = V_max	Bisection on V(y) - V_max = 0
Max Shear Stress	τ_max (N/m²)	Depth from y = τ_max / (γ·S), then flow	Direct (bed stress is linear in y)

Enter Depth is useful when the flow depth is known from upstream controls or a stage measurement. Enter Flow is the standard sizing case: given a design discharge from a Rational, SCS, or SDF calculation, find the normal depth. Max Velocity and Max Shear Stress are useful for permissible-velocity or permissible-shear sizing — enter the allowable limit and the tool returns the largest depth (and flow) that stays within it.

Hydraulic curves

After calculating results, the tool generates hydraulic curves at 50 depth increments from zero to 2.5 × the solved depth. You can configure the axes to plot any combination of y, A, P, R, T, D_h, V, Q, Fr, τ_avg, τ_max, specific energy, or Froude number.

• X-axis: choose any hydraulic parameter.
• Y-axis (left): select one or more parameters for the primary axis.
• Y-axis (right): add parameters on a secondary axis with a different scale.

Common combinations include:

• Stage–discharge (y vs Q) — the rating curve for the channel, used for hydrograph routing and gauge calibration.
• Depth–velocity (y vs V) — shows where velocity limits are reached.
• Flow–Froude (Q vs Fr) — reveals the discharge at which the channel transitions between regimes.

Charts can be downloaded as PNG images and the underlying data exported as CSV for use in reports and spreadsheets.

Worked example — sizing a trapezoidal drain

Given

• Design flow: Q = 8.0 m³/s (1:50 year peak from a Rational Method calculation)
• Channel shape: trapezoidal, bottom width B = 2.0 m, side slopes Z₁ = Z₂ = 1.5 (1.5H:1V)
• Lining: float-finished concrete, n = 0.015
• Longitudinal slope: S = 0.004 m/m
• Find: normal depth, velocity, Froude number, critical depth, boundary shear stress, and recommended freeboard.

Step 1 — Solve for normal depth. Use Enter Flow mode. The tool iterates Manning’s equation:

$$\tfrac{1}{0.015}\,A(y_n)\,R(y_n)^{2/3}\,(0.004)^{1/2} \;=\; 8.0$$

with A(y) = 2.0 y + 1.5 y² and P(y) = 2.0 + 2 y √(1+1.5²) = 2.0 + 3.606 y. Bisection converges to y_n ≈ 1.12 m.

Step 2 — Compute geometric properties at y_n.

$$\begin{aligned} A &= 2.0(1.12) + 1.5(1.12)^{2} = 4.12\ \mathrm{m^{2}} \\ P &= 2.0 + 3.606(1.12) = 6.04\ \mathrm{m} \\ R &= A/P = 0.683\ \mathrm{m} \\ T &= 2.0 + 2(1.5)(1.12) = 5.36\ \mathrm{m} \\ D_h &= A/T = 0.769\ \mathrm{m} \end{aligned}$$

Step 3 — Velocity and Froude number.

$$V = Q/A = 8.0/4.12 = 1.94\ \mathrm{m/s} \qquad Fr = \frac{V}{\sqrt{g\,D_h}} = \frac{1.94}{\sqrt{9.81 \times 0.769}} = 0.71$$

Fr = 0.71 — comfortably subcritical.

Step 4 — Critical depth. Solve Q² T / (g A³) = 1 iteratively for the same section → y_c ≈ 0.86 m. Since y_n > y_c, the slope is mild and normal flow is subcritical. ✓

Step 5 — Boundary shear stress.

$$\tau_\text{avg} = \gamma R S = 9810 \times 0.683 \times 0.004 = 26.8\ \mathrm{N/m^{2}} \qquad \tau_\text{max} = \gamma y S = 9810 \times 1.12 \times 0.004 = 43.9\ \mathrm{N/m^{2}}$$

Concrete lining is effectively unerodible at these values (permissible τ > 1000 N/m²). ✓

Step 6 — Freeboard. Q = 8.0 m³/s falls in the 1.5 – 85 m³/s band:

$$f \;=\; 0.45 + \frac{V^{2}}{2g} \;=\; 0.45 + \frac{1.94^{2}}{19.62} \;=\; 0.45 + 0.19 \;=\; 0.64\ \mathrm{m}$$

Step 7 — Total channel depth. H_total = y_n + f = 1.12 + 0.64 = 1.76 m. Round up to 1.80 m for construction. The top width at the lip is T_top = B + (Z₁+Z₂)·H_total = 2.0 + 3.0×1.80 = 7.4 m.

The channel operates comfortably in the subcritical regime with a healthy margin below critical depth, well within permissible shear stress, and with adequate freeboard for waves and uncertainty in the design flow.

Limitations

• Uniform flow only. The tool assumes the friction slope equals the bed slope. Transitions, controls, backwater, and drawdown require GVF analysis — use the GVF Profiles tool instead.
• Prismatic section assumption. All geometries are assumed constant along the reach. Use the custom cross-section input for a single representative section; non-prismatic channels require a multi-section solver.
• Single Manning’s n. One roughness value is applied to the entire wetted perimeter. Composite-n situations (main channel + vegetated overbank) must be handled externally.
• Steady flow. No unsteady-flow effects (surging, waves, pressure transients) are modelled.
• Fully-turbulent regime. Manning’s equation presumes fully-turbulent flow (Re > 2000). Very small, slow flows (laminar or transitional) may deviate from the R^(2/3) scaling.
• Free-surface flow only. Circular-pipe results apply up to the capacity depth y/D ≈ 0.938. For pressurised (surcharged) pipe flow, use a pressurised-flow calculator such as Darcy-Weisbach or Hazen-Williams.

References

• Chow, V.T. (1959). Open-Channel Hydraulics. McGraw-Hill, New York. The canonical reference — Chapters 5 (uniform flow), 6 (design of channels), and 7 (specific energy and critical flow) underpin the methods implemented here.
• SANRAL. (2013). Drainage Manual (6th ed.). South African National Roads Agency, Pretoria. Chapter 7 — Surface Drainage: permissible velocities, freeboard guidelines, and lining design for South African practice.
• USACE. (1994). Hydraulic Design of Flood Control Channels. Engineer Manual EM 1110-2-1601. U.S. Army Corps of Engineers, Washington, D.C. Authoritative guidance on Manning’s n selection, super-elevation on curves, and permissible shear stresses.
• Henderson, F.M. (1966). Open Channel Flow. Macmillan, New York. Classical treatment of specific energy, alternate depths, and flow regimes.
• USBR. (1987). Design of Small Canals and Related Structures. U.S. Bureau of Reclamation, Denver. Source for the freeboard band thresholds used in the recommended-freeboard calculation.

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Open Channel Analysis