Free normal depth calculator. Solve uniform-flow depth for circular pipes and rectangular, trapezoidal, or triangular channels with Manning’s equation. Returns depth, velocity, hydraulic radius, and Froude number.
Quick answer
Normal depth (yn) is the constant depth of uniform flow in a prismatic channel or partially full pipe. It occurs when gravity along the bed slope exactly balances boundary friction, so the water-surface, energy, and bed slopes are parallel.
Manning’s equation solved for the unknown depth
Q = (k/n) A(yn) R(yn)2/3 S1/2
Use k = 1.486 with cfs and feet, or k = 1.0 with m3/s and metres. The solver supports circular, rectangular, trapezoidal, and triangular sections and reports depth, velocity, hydraulic radius, and Froude number.
Normal depth is the depth at which uniform flow occurs in an open channel. At normal depth, the energy slope equals the channel bed slope, meaning gravitational forces are balanced by frictional resistance.
Manning's equation for uniform flow:
This calculator iteratively solves for the depth that satisfies Manning's equation for the given discharge.
| Material | n Value |
|---|---|
| Concrete pipe | 0.012 - 0.015 |
| Corrugated metal pipe | 0.022 - 0.027 |
| HDPE pipe | 0.010 - 0.012 |
| Concrete channel | 0.013 - 0.017 |
| Earth channel (clean) | 0.022 - 0.033 |
| Grass-lined channel | 0.030 - 0.050 |
Source: Chow (1959), HEC-22 Table 7-1.
For educational purposes only. Not a substitute for professional engineering judgment.
Interactive model
Adjust the interactive channel view to see how section geometry, slope, and roughness shape uniform-flow depth.
Open full model (opens in a new tab)Normal depth is the depth at which uniform flow occurs: the energy slope equals the channel bed slope, so gravity is balanced by friction and the depth stays constant along a prismatic channel. It is governed by Manning’s equation for steady, uniform open-channel flow:
The constant k is 1.486 in US customary units and 1.0 in SI units. Both the flow area A and the hydraulic radius R = A / P change with depth, so the equation cannot be rearranged for depth directly. The solver searches for the depth yn at which the computed Q equals the target discharge, using Newton–Raphson iteration with a bisection fallback (up to 100 iterations).
The result is then checked for flow regime with the Froude number:
Variable definitions
These examples show how slope, section geometry, and Manning’s n change the uniform-flow depth. Values are rounded from the same solver used above.
A 450 mm circular pipe carries Q = 0.100 m3/s on a 0.6% slope. Changing only roughness shows the hydraulic cost of corrugations.
| Pipe | Manning’s n | Normal depth | Velocity | Froude number |
|---|---|---|---|---|
| Concrete | 0.013 | 0.212 m | 1.354 m/s | 1.066 |
| Corrugated metal | 0.024 | 0.315 m | 0.842 m/s | 0.501 |
Q = 100 cfs, b = 10 ft, S = 0.001, and n = 0.025 gives yn ≈ 3.35 ft with the Manning solver used above.
Q = 100 cfs, b = 10 ft, z = 2H:1V, S = 0.001, and n = 0.025 gives yn = 2.459 ft, V = 2.726 ft/s, and Fr = 0.353.
Manning’s equation needs the flow area, wetted perimeter, and top width as functions of depth. These are the exact relationships this calculator uses for each shape (b = bottom width, y = depth, z = side slope as horizontal:vertical, D = pipe diameter).
| Shape | Flow area, A | Wetted perimeter, P | Top width, T |
|---|---|---|---|
| Rectangular | b·y | b + 2y | b |
| Trapezoidal | (b + z·y)·y | b + 2y·√(1 + z²) | b + 2z·y |
| Triangular (V) | z·y² | 2y·√(1 + z²) | 2z·y |
| Circular (partial) | (D²/8)(θ − sinθ) | (D/2)·θ | D·sin(θ/2) |
Symmetric side slopes assumed for trapezoidal and triangular sections. For circular pipes, θ is the central angle (radians) subtended by the water surface and the hydraulic radius is R = A / P. Geometry per Chow (1959), Open-Channel Hydraulics.
The roughness coefficient n has a large, nonlinear effect on the result. For example, in a wide rectangular channel, depth scales approximately with n3/5, so doubling n raises depth by about 52%; the exact response depends on section geometry. Use a representative value for the channel lining or pipe material.
| Material / lining | Manning’s n |
|---|---|
| HDPE pipe (smooth) | 0.010 – 0.012 |
| Concrete pipe | 0.012 – 0.015 |
| Concrete-lined channel | 0.013 – 0.017 |
| Corrugated metal pipe | 0.022 – 0.027 |
| Earth channel (clean, straight) | 0.022 – 0.033 |
| Grass-lined channel | 0.030 – 0.050 |
Source: Chow (1959), Open-Channel Hydraulics; FHWA HEC-22, Urban Drainage Design Manual (3rd ed., 2009).
Normal depth (yₙ) is the flow depth at which uniform flow occurs in a prismatic channel or pipe — the depth where the water-surface slope, energy slope, and channel bed slope are all parallel. At that depth the gravitational driving force exactly balances the boundary friction, so the depth no longer changes along the channel. It is found by solving Manning’s equation for the depth that conveys the design discharge.
There is no closed-form solution because flow area (A) and hydraulic radius (R) both depend on the unknown depth. This calculator solves Manning’s equation Q = (k/n)·A·R^(2/3)·S^(1/2) iteratively: it uses the Newton–Raphson method (fast convergence) and automatically falls back to bisection (robust) if needed, refining the depth until the computed discharge matches your target Q to a tight tolerance.
Normal depth depends on slope and roughness and is the depth of uniform flow for a given discharge. Critical depth depends only on discharge and channel geometry (the point of minimum specific energy, Froude number = 1). When normal depth is greater than critical depth the channel is mild and flow is subcritical (Fr < 1); when normal depth is less than critical depth the slope is steep and flow is supercritical (Fr > 1). This tool reports the Froude number and flow type so you can tell which regime you are in.
On a very flat slope a channel may not be able to carry the target discharge at any reasonable depth, or for circular pipes the depth can approach the crown where the relationship between depth and capacity becomes non-unique. If the solver does not converge it returns a warning rather than a silent wrong answer — try a steeper slope, a larger section, or a different starting depth.
Iteratively solves Manning's equation to find the normal depth — the depth at which uniform flow occurs in an open channel or pipe for a given discharge, slope, and roughness.
Normal depth is very sensitive to Manning's n and channel slope. Small errors in these inputs produce significant changes in depth. Verify roughness values against field conditions.
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Last verified: February 2026