HomeCalculatorsPhysicsFluid MechanicsWeir Discharge Calculator
Physics calculator

Weir Discharge Calculator

Compute thin-plate weir discharge. Rectangular Q=(2/3) C_d L √(2g) H^{3/2}. V-notch Q=(8/15) C_d √(2g) tan(θ/2) H^{5/2}. Runs locally. Not Manning, not Q=Av, not Darcy, and not an orifice.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Thin-plate rectangular and triangular free-flow weir discharge, + O3 mpmath tabulated Q.
Input interpretation
Enter values to calculate.
Result
—
Verified scope
Thin-plate rectangular and triangular free-flow weir discharge, + O3 mpmath tabulated Q.
Assurance
Engineering
Declared partition coverage
PASS · 5/5 declared partitions (rectangular, triangular, alias, awkward, invalid-domain) · Matrix
Deferred
Not submerged H2, not orifice, not Manning, not Darcy, not Parshall/sluice/ogee/side.
Numerical scope
O2: thin-plate weir vs a separate-module identity (≤2 ULP). Not submerged H2, not orifice, not Manning, and not Darcy. ≤2 ULP vs O3 applies only to the published tabulated weir vectors. Not submerged H2, not orifice, not Manning, and not Darcy.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Thin-plate weir discharge or head. Rectangular and 90° V-notch (θ adjustable).
Scope
Thin-plate weir; free nappe
Verification
Engine tested · Source checked · v1.0.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Thin-plate rectangular and triangular free-flow weir discharge, + O3 mpmath tabulated Q.· 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 cb016aee9f08
Legacy regression
15/15 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 · 7/7 CVP boundary · 10/10 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 · 9 legacy boundary · legacy regression suite · 4/4 oracle-backed golden · 10/10 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.

RectangularQ = (2/3) C_d L √(2g) H^{3/2}
Triangular / V-notchQ = (8/15) C_d √(2g) tan(θ/2) H^{5/2}
iSI g = 9.80665 m/s² unless given. Default C_d = 0.62 (rectangular) or 0.58 (90° V-notch). Francis 3.33 L H^{3/2}, Kindsvater–Carter contractions, Cipolletti, Manning, Q=Av, Darcy, and orifice plates are out of scope.

How to use

1

Choose the weir shape

Rectangular (suppressed thin-plate) or triangular V-notch. This is not a Manning or orifice page.

2

Enter head H or discharge Q

Give one, not both. H is the upstream head above the crest.

3

Enter L or θ

Rectangular needs crest length L. V-notch uses θ (default 90°).

4

Optional C_d

Leave blank for 0.62 / 0.58. C_d must be in (0, 1].

Example calculations

Common configurations with formula and result.

ϟ

Rectangular 0.2 m head

L=1 m, H=0.2 m, default C_d=0.62

Q = (2/3)(0.62)(1)√(2g) (0.2)^{3/2}
Q in m³/s
ϟ

90° V-notch

θ=90°, H=0.2 m, default C_d=0.58

Q = (8/15)(0.58)√(2g) tan(45°) (0.2)^{5/2}
Q in m³/s

Weir Discharge calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Thin-plate weir. Rectangular: Q=(2/3) C_d L √(2g) H^{3/2}. Triangular: Q=(8/15) C_d √(2g) tan(θ/2) H^{5/2}.
What it calculates
Thin-plate weir discharge or head. Rectangular and 90° V-notch (θ adjustable).
Inputs
  • mode?
  • H | Q
  • L?
  • theta_deg?
  • Cd?
  • g?
Outputs
  • Q
  • H
  • Cd
  • g
  • L?
  • theta_deg?
Formula
Q = (2/3) C_d L √(2g) H^{3/2} | Q = (8/15) C_d √(2g) tan(θ/2) H^{5/2}
Assumptions
  • Thin-plate weir; free nappe
  • SI g=9.80665 unless given
  • Rectangular: Q=(2/3) C_d L √(2g) H^{3/2}; not Francis contractions
  • Triangular: Q=(8/15) C_d √(2g) tan(θ/2) H^{5/2}
  • Not Manning, not Q=Av, not Darcy, not an orifice
Units
  • SI; L and H in m; Q in m³/s; θ in degrees; g in m/s²
Boundary conditions
  • missing H or Q → MISSING_REQUIRED_INPUT
  • rectangular missing L → MISSING_REQUIRED_INPUT
  • H, Q, L, Cd, θ, or g ≤ 0 → VALUE_MUST_BE_POSITIVE
  • H and Q together, or θ_deg and θ_rad, or H2 → MIXED_INPUT_ENCODING
  • C_d > 1 → VALUE_OUT_OF_RANGE
  • Francis, Cipolletti, Manning, or orifice as mode → INVALID_MODE
Example
L=1 H=0.2 → Q=(2/3)(0.62)(1)√(2g) H^{3/2}
Validation cases

3 published on this page · 15/15 tests · Production surface contract 5/5 · View evidence

  • L=1 H=0.2 → Q = (2/3) 0.62 L √(2g) H^{3/2}
  • mode=triangular H=0.2 → Q = (8/15) 0.58 √(2g) tan(θ/2) H^{5/2}
  • L=1 H=0.2 Cd=3.33 → VALUE_OUT_OF_RANGE
Sources
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Compute thin-plate weir discharge from upstream head.

Supported and not supported

Supported — rectangular and triangular (V-notch) thin-plate weirs · invert H from Q · API physics.fluid.weir

Not supported — Manning, Q=Av identities, Darcy–Weisbach, orifice plates, Francis 3.33 L H^{3/2}, Kindsvater–Carter contractions, Cipolletti, submerged nappe (that is /calc/physics/submerged-weir), broad-crested weir

Agent / API notes

Capability id: physics.fluid.weir · tool id: weir-discharge · pin 1.0.0.

{ "mode": "rectangular", "L": 1, "H": 0.2 }

V-notch: { "mode": "triangular", "H": 0.2 } (θ defaults to 90°). Invert: { "L": 1, "Q": 0.1638 }. Errors: MISSING_REQUIRED_INPUT, INVALID_NUMBER, VALUE_MUST_BE_POSITIVE, VALUE_OUT_OF_RANGE, MIXED_INPUT_ENCODING, INVALID_MODE.

Calculator URL stays /calc/physics/weir-discharge. 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 the Manning formula?

No. Manning is uniform open-channel flow from n, A, R, and S at /calc/physics/manning-formula. A weir is a thin-plate overflow.

Is this a broad-crested weir?

No. Rectangular broad-crested Q=C_d (2/3)^{3/2} b √g H^{3/2} is /calc/physics/broad-crested-weir. Sending kind=broad-crested here is MIXED_INPUT_ENCODING.

Is this a Parshall flume?

No. USBR 1–8 ft Parshall Q from throat W and gage Ha is /calc/physics/parshall-flume. Sending kind=parshall here is MIXED_INPUT_ENCODING.

Is this a cutthroat flume?

No. Skogerboe cutthroat Q from W, independent L_f, and Ha is /calc/physics/cutthroat-flume. Sending kind=cutthroat here is MIXED_INPUT_ENCODING.

Is this a sluice gate?

No. A free-flow undershot sluice with Swamee C_d from a/y1 is /calc/physics/sluice-gate. Sending kind=sluice here is MIXED_INPUT_ENCODING.

Is this an ogee spillway?

No. USBR high-overflow ogee Q from L, H, and design head Hd is /calc/physics/ogee-spillway. Sending kind=ogee here is MIXED_INPUT_ENCODING.

Is this a side weir?

No. Borghei 1999 lateral overflow from B, y1, p, and Q1 is /calc/physics/side-weir. Sending kind=side here is MIXED_INPUT_ENCODING.

Is this a submerged weir?

No. Villemonte drowned nappe from H1 and H2 is /calc/physics/submerged-weir. Sending H2 here is MIXED_INPUT_ENCODING.

Is this Q = A·v?

No. Volumetric flow from a given area and velocity is /calc/physics/flow-rate.

Can I use the US Francis formula 3.33 L H^{3/2}?

No. That customary coefficient is out of scope. C_d must be in (0, 1]. Contractions (Kindsvater–Carter) are also out of scope.

Is this an orifice?

No. A hole under head or a pipe orifice plate is /calc/physics/orifice-discharge. A weir is a thin-plate overflow.

Where does this run?

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