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Physics calculator

Manning Formula Calculator

Compute SI Manning discharge Q = (1/n) A R^{2/3} S^{1/2}. Runs locally. Not Q=Av, not Darcy–Weisbach, and not a weir.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · SI Manning Q = (1/n) A R^{2/3} S^{1/2}, plus velocity, n, and S solves, + O3 mpmath tabulated discharge.
Input interpretation
Enter values to calculate.
Result
—
Verified scope
SI Manning Q = (1/n) A R^{2/3} S^{1/2}, plus velocity, n, and S solves, + O3 mpmath tabulated discharge.
Assurance
Engineering
Declared partition coverage
PASS · 5/5 declared partitions (discharge, solve, alias, awkward, invalid-domain) · Matrix
Deferred
Not Darcy f, not Chezy C, not Hazen–Williams, not Q=Av, and not normal-depth yn.
Numerical scope
O2: SI Manning vs a separate-module identity (≤2 ULP). Not Darcy f, not Chezy C, not Hazen–Williams, not Q=Av, and not normal-depth yn. ≤2 ULP vs O3 applies only to the published tabulated Manning vectors. Not Darcy f, not Chezy C, not Hazen–Williams, not Q=Av, and not normal-depth yn.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
SI Manning open-channel discharge and mean velocity. Invert for n or S.
Scope
SI Manning k=1
Verification
Engine tested · Source checked · v1.0.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · SI Manning Q = (1/n) A R^{2/3} S^{1/2}, plus velocity, n, and S solves, + O3 mpmath tabulated discharge.· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-26.o2-o3
Verification revision
2026-09-26.o2-o3 · 1/1 property · digest 59bd904ca406
Legacy regression
13/13 tests · Production surface contract 5/5
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
4/4 golden · 6/6 CVP boundary · 8/8 invalid · 1/1 property · 1/1 metamorphic · 8/8 O3 · 2/2 cross-interface · 5/5 CVP contract · Manifest
Sources
Sources
Evidence
1 legacy golden · 7 legacy boundary · legacy regression suite · 4/4 oracle-backed golden · 8/8 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.0 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

DischargeQ = (1/n) A R^{2/3} S^{1/2}
Mean velocityv = (1/n) R^{2/3} S^{1/2}
iSI only (k=1). US customary 1.486 is out of scope. This page is not Q=Av, not Darcy–Weisbach, and not a weir.

How to use

1

Enter Manning n

n is the empirical roughness. It must be greater than zero.

2

Enter area A and hydraulic radius R

R may be given directly, or as A/P. Do not send R and P together.

3

Enter slope S

S is the friction slope (rise/run). It must be greater than zero.

4

Read Q and v

v = (1/n) R^{2/3} S^{1/2}. Q = A v. Invert for n or S on the other tabs.

Example calculations

Common configurations with formula and result.

ϟ

Concrete channel

n=0.013, A=1 m², R=0.5 m, S=0.001

v = (1/0.013) (0.5)^{2/3} √0.001
Q = A v
ϟ

From wetted perimeter

n=0.013, A=1 m², P=2 m, S=0.001

R = A/P = 0.5 m
same Q as R=0.5 m

Manning Formula calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
SI Manning: Q = (1/n) A R^{2/3} S^{1/2}. k = 1. Uniform open-channel flow. R = A/P.
What it calculates
SI Manning open-channel discharge and mean velocity. Invert for n or S.
Inputs
  • mode?
  • n
  • A
  • R | P
  • S
  • Q?
Outputs
  • Q?
  • v
  • n
  • A?
  • R
  • S
  • k
Formula
Q = (1/n) A R^{2/3} S^{1/2}
Assumptions
  • SI Manning k=1
  • Uniform open-channel flow
  • R = A/P
  • Not Q=Av identities
  • Not Darcy–Weisbach
  • Not a weir
Units
  • SI; A in m²; R and P in m; Q in m³/s; S dimensionless
Boundary conditions
  • missing n, A, R/P, or S when required → MISSING_REQUIRED_INPUT
  • n, A, R, P, S, or Q ≤ 0 → VALUE_MUST_BE_POSITIVE
  • R and P together → MIXED_INPUT_ENCODING
  • k given and k ≠ 1 → INVALID_MODE
  • weir is not a mode on this page
Example
n=0.013 A=1 R=0.5 S=0.001 → Q=(1/n) A R^{2/3} S^{1/2}
Validation cases

3 published on this page · 13/13 tests · Production surface contract 5/5 · View evidence

  • n=0.013 A=1 R=0.5 S=0.001 → Q = A (1/n) R^{2/3} S^{1/2}
  • n=0.013 A=1 P=2 S=0.001 → R=0.5
  • n=0 A=1 R=0.5 S=0.001 → VALUE_MUST_BE_POSITIVE
Sources
  • ISO 80000-4 — Mechanics — Volume flow rate and hydraulic radius
    Supports: Q is volume flow; hydraulic radius R = A/P
  • Chow, V.T. Open-Channel Hydraulics — Manning formula
    Supports: SI form Q = (1/n) A R^{2/3} S^{1/2} for uniform flow
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Compute open-channel discharge from the SI Manning formula.

Supported and not supported

Supported — Q, v, n, or S from the SI Manning formula · API physics.fluid.manning

Not supported — Q=Av identities, Darcy–Weisbach, Colebrook, weirs, US customary k=1.486, compound channels

Agent / API notes

Capability id: physics.fluid.manning · tool id: manning-formula · pin 1.0.0.

{ "mode": "discharge", "n": 0.013, "A": 1, "R": 0.5, "S": 0.001 }

Alternate hydraulic-radius encoding: { "n": 0.013, "A": 1, "P": 2, "S": 0.001 }. Modes: discharge (default), velocity, n, slope. Errors: MISSING_REQUIRED_INPUT, INVALID_NUMBER, VALUE_MUST_BE_POSITIVE, MIXED_INPUT_ENCODING, INVALID_MODE.

Calculator URL stays /calc/physics/manning-formula. There is no /calc/fluid.

Other calculators in this family: Alternate Depth Calculator, Bernoulli Equation Calculator, Broad-Crested Weir Calculator, Cavitation Number Calculator, Chezy Formula Calculator, Colebrook Friction Factor Calculator, Concentric Reducer Calculator, Critical Depth Calculator . Explore all Fluid Mechanics.

→ Alternate Depth Calculator Compute both rectangular depths at the same specific energy from E, Q, and width b. Runs locally. Not specific energy from y+v; not specific force M; not hydraulic jump y2; not critical depth yc as the product. › → Bernoulli Equation Calculator Incompressible Bernoulli along a streamline: solve P2, v2, or h2. Runs locally in your browser. › → Broad-Crested Weir Calculator Compute rectangular broad-crested weir discharge Q = C_d (2/3)^{3/2} b √g H^{3/2}. Default C_d=0.85. Runs locally. Not a thin-plate weir, not Manning, not Q=Av, and not an orifice. › → Cavitation Number Calculator Compute the cavitation number σ = (p − p_v) / (½ ρ v²). Runs locally. Not NPSH, not Thoma NPSH/H, not Weber, and not Euler Eu=Δp/(ρ v²). › → Chezy Formula Calculator Compute open-channel mean velocity V = C √(R S). Runs locally. Not Manning n, not Q=Av, not Hazen–Williams, and not Darcy. › → Colebrook Friction Factor Calculator Solve the Colebrook–White equation for Darcy friction factor f from Re and ε/D. Turbulent pipes Re≥4000. Runs locally. Not laminar 64/Re and not a Moody-chart lookup. › → Concentric Reducer Calculator Compute concentric-reducer loss h = K v₂² / (2g) with K = 0.5 (1 − A₂/A₁) referenced to the downstream speed. Runs locally. Not sudden expansion, not gradual contraction α, not vena-contracta C_c, not catalog K, and not Q=Av. › → Critical Depth Calculator Compute rectangular critical depth yc = (q²/g)^{1/3}. Runs locally. Not a Froude wrap, not specific energy, not a hydraulic jump, and not Q=Av. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this Q = A·v?

No. Volumetric flow from a given area and velocity is /calc/physics/flow-rate. This page finds v from Manning n, R, and S.

Is this Darcy–Weisbach?

No. Pipe head loss from a given f is /calc/physics/darcy-weisbach. Manning is uniform open-channel flow.

Is this a weir?

No. Thin-plate weir discharge is /calc/physics/weir-discharge. This page is SI Manning only.

Can I use US customary k = 1.486?

No. This page is SI only (k=1). Sending k other than 1 is rejected.

Where does this run?

Locally in the browser by default. REST and MCP call the same fluid-engine Manning wrapper.