What This Solves
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.
Best Used When
- You need to find the equilibrium flow depth in a long channel or pipe at a given discharge
- You are checking whether a channel will overtop its banks at the design flow
- You need normal depth as a boundary condition for water surface profile calculations
Do NOT Use When
- You need critical depth rather than normal depth — Use Critical Depth Calculator
- You already know the depth and need to calculate the flow rate — Use Manning's Channel Calculator
Key Assumptions
- Uniform flow conditions exist (depth and velocity are constant along the channel)
- The channel has a constant slope, roughness, and cross-section
- The energy grade line slope equals the channel bed slope
- Flow is fully turbulent (Manning's equation is applicable)
Input Quality Notes
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.
Solve for normal depth — the depth of uniform flow — in circular pipes and rectangular, trapezoidal, or triangular channels. Enter the design discharge, slope, roughness, and geometry, and the calculator iterates Manning’s equation to return the depth along with velocity, hydraulic radius, top width, and the Froude number.
Normal Depth Overview
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:
- US Units: Q = (1.486/n) * A * R2/3 * S1/2
- SI Units: Q = (1/n) * A * R2/3 * S1/2
This calculator iteratively solves for the depth that satisfies Manning's equation for the given discharge.
Typical Manning's n Values
| 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.
How normal depth is calculated
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:
- US customary: Q = (1.486 / n) · A · R2/3 · S1/2
- SI (metric): Q = (1 / n) · A · R2/3 · S1/2
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:
- Fr = V / √(g · Dh), where Dh = A / T (hydraulic depth) and g = 32.174 ft/s² (9.81 m/s²)
- Fr < 1 — subcritical (mild slope, deep slow flow)
- Fr = 1 — critical flow
- Fr > 1 — supercritical (steep slope, shallow fast flow)
Variable definitions
- Q — discharge (cfs or m³/s)
- n — Manning’s roughness coefficient (dimensionless)
- A — cross-sectional flow area at depth y
- R — hydraulic radius = A / P (wetted area ÷ wetted perimeter)
- P — wetted perimeter (length of channel boundary in contact with water)
- S — channel bed slope (ft/ft or m/m)
- T — top width of the water surface
- yn — normal depth (the unknown being solved)
Cross-section geometry equations
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.
Typical Manning’s n values
The roughness coefficient n has a large effect on the result — doubling n roughly doubles the required depth for the same discharge. 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).
Frequently asked questions
What is normal depth?
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.
How is normal depth calculated?
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.
What is the difference between normal depth and critical depth?
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.
Why does a solution sometimes fail to converge?
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.
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Last verified: February 2026