Open Channel Flow Calculator (Manning's Equation)
Use this calculator to find the water velocity and volumetric discharge in any open channel using Manning's equation. Choose a channel shape (rectangular, trapezoidal, triangular, or circular), set the slope, depth, and lining roughness, and the results update instantly. You also get the hydraulic radius, Froude number, and critical depth so you can tell at a glance whether the flow is subcritical or supercritical. Switch between metric and imperial units at any time.
What is Manning's equation?
Manning's equation is the most widely used formula for computing uniform flow velocity and discharge in open channels. Published by the Irish engineer Robert Manning in 1889, it relates the average cross-sectional velocity to the channel's hydraulic radius, slope, and surface roughness: V = (k/n) * R^(2/3) * S^(1/2), where k = 1.0 in SI units and 1.486 in imperial units. Once velocity is known, discharge Q = V * A, where A is the cross-sectional flow area. The key assumption is that flow is steady (not changing over time) and uniform (water depth is constant along the channel), which is a reasonable approximation for long, prismatic channels at normal design conditions.
Channel geometry: area, wetted perimeter, and hydraulic radius
Three geometric properties drive Manning's equation. The flow area A is the cross-sectional area of water in the channel. The wetted perimeter P is the length of the channel boundary actually in contact with the water (the water surface is not included). The hydraulic radius R = A / P represents the average depth of flow weighted by the perimeter, and it is the primary geometric variable in Manning's formula. For rectangular channels, A = b * y and P = b + 2y. For trapezoidal channels, A = (b + z*y)*y and P = b + 2*y*sqrt(1+z^2). Circular pipes use trigonometric expressions for the arc subtended by the water surface. The hydraulic depth D_h = A / T (where T is the top water-surface width) is used separately to compute the Froude number.
Froude number and flow regime
The Froude number Fr = V / sqrt(g * D_h) classifies the flow regime. When Fr < 1 the flow is subcritical (tranquil or streaming): gravity dominates, surface waves can travel upstream, and downstream conditions control the flow. When Fr > 1 the flow is supercritical (rapid or shooting): inertia dominates, waves cannot propagate upstream, and upstream conditions govern. At Fr = 1 the flow is critical, which is an unstable condition sensitive to small perturbations. Most drainage channels and irrigation canals are designed to operate in the subcritical regime. Spillways, steep chutes, and steep natural streams often develop supercritical flow, which can transition back to subcritical through a hydraulic jump that dissipates energy.
Critical depth and specific energy
For a given discharge Q and channel geometry, the critical depth y_c is the water depth at which the specific energy E = y + V^2/(2g) is at its minimum. It corresponds to Fr = 1. Critical depth is found by iterating the condition Q^2 * T / (g * A^3) = 1, which this calculator solves numerically. Specific energy is the sum of the depth (potential energy head relative to the channel bed) and the velocity head V^2/(2g). For subcritical flows the depth is above critical and the velocity head is small; for supercritical flows the depth is below critical and the velocity head is large. The depth-discharge chart in this calculator traces the rating curve, showing how Q and V vary with y for the configured channel - a useful tool for checking if the channel can carry a design flood.
Common Manning's n values
| Channel lining / material | Typical n range | Design n |
|---|---|---|
| Neat cement / planed wood | 0.010-0.013 | 0.011 |
| Mortar / trowel finish concrete | 0.011-0.015 | 0.013 |
| Formed concrete (good finish) | 0.013-0.017 | 0.015 |
| Smooth asphalt | 0.012-0.016 | 0.013 |
| Corrugated metal pipe | 0.022-0.027 | 0.024 |
| Earth channel, clean straight | 0.018-0.025 | 0.022 |
| Earth with gravel or sand bottom | 0.022-0.030 | 0.025 |
| Riprap / rock lining | 0.030-0.045 | 0.035 |
| Grass-lined, short (< 150 mm) | 0.025-0.040 | 0.033 |
| Grass-lined, tall / dense | 0.040-0.100 | 0.050 |
| Natural stream, clean and straight | 0.025-0.033 | 0.030 |
| Natural stream with pools / bends | 0.033-0.045 | 0.040 |
Representative roughness coefficients for channel design. Select a wider range as appropriate for condition uncertainty.
Frequently asked questions
What is Manning's roughness coefficient n?
Manning's n is a dimensionless empirical coefficient that accounts for the resistance to flow caused by the channel lining. Lower values (around 0.011 for smooth concrete) indicate a smooth surface with little friction; higher values (0.050 or more for dense grass or natural streams with obstructions) indicate rough surfaces with high resistance. Published tables provide n ranges for common lining types. In practice an engineer selects a value in the middle-to-upper end of the range to be conservative. Incorrect n is one of the most common sources of error in open-channel hydraulic design.
What is the difference between subcritical and supercritical flow?
Subcritical flow (Froude number < 1) is tranquil and gravity-dominated. Downstream conditions, such as a weir or tail water level, control the depth. Supercritical flow (Froude number > 1) is rapid and inertia-dominated; only upstream conditions matter. In practice, most lined drainage channels and irrigation canals are designed to be subcritical for stability. Steep chutes and spillways are supercritical. When supercritical flow hits a mild slope or an obstruction it transitions to subcritical through a hydraulic jump, which releases energy as turbulence.
How is the discharge (Q) in an open channel calculated?
Using Manning's equation: (1) compute the flow area A and wetted perimeter P from the channel cross-section and water depth; (2) compute the hydraulic radius R = A/P; (3) apply Manning's velocity formula V = (1/n) * R^(2/3) * S^(1/2) (in SI); (4) multiply by the area to get Q = V * A. The calculator performs all four steps instantly as you type. In imperial units, the leading coefficient is 1.486 instead of 1.0.
How do I find the normal depth for a given discharge?
Normal depth is the equilibrium water depth at which a given discharge flows uniformly for a specified slope and roughness. Because Manning's equation cannot be solved directly for depth when Q is known, normal depth is found by iteration: guess a depth, compute Q, and adjust until computed Q matches the target. Most open-channel design software and spreadsheets do this numerically. A future version of this calculator will include a normal-depth solver. For now, adjust the depth input until the displayed discharge matches your target.
When does Manning's equation not apply?
Manning's equation assumes steady, uniform flow in a prismatic channel (constant cross-section and slope). It does not apply to rapidly varied flow such as in a hydraulic jump, at channel transitions (contractions, expansions), at weirs or gates, or in unsteady (time-varying) floods. For those cases, energy or momentum equations, or full 1D/2D hydrodynamic models, are needed. Even for uniform flow, Manning's n is empirical and carries uncertainty, so results should be treated as engineering estimates.
What channel shape is most hydraulically efficient?
A channel is most hydraulically efficient when it conveys the maximum discharge for a given cross-sectional area, which occurs when the wetted perimeter is minimized for that area. In theory the semi-circle has the smallest wetted perimeter for any area and is thus the most efficient shape. Among practical prismatic channels: for rectangular channels the most efficient section has a width equal to twice the depth (b = 2y); for trapezoidal channels it is a half-hexagon (side slopes at 60 degrees, giving z = 1/sqrt(3) approximately 0.577). These results emerge directly from Manning's equation and are commonly used as starting points in canal design.