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Physics calculator

Submerged Weir Calculator

Compute Villemonte rectangular submerged-weir discharge from crest L, upstream head H1, and downstream head H2. Qs=Qf [1−(H2/H1)^{1.5}]^{0.385}. Runs locally. Not free-flow thin-plate H only, not an ogee, not a side weir, not Manning, and not Q=Av.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Villemonte 1947 rectangular submerged weir Qs=Qf [1−(H2/H1)^{1.5}]^{0.385} with Qf=(2/3) Cd L √(2g) H1^{3/2}, + O3 mpmath tabulated Q.
Input interpretation
Enter values to calculate.
Result
—
Verified scope
Villemonte 1947 rectangular submerged weir Qs=Qf [1−(H2/H1)^{1.5}]^{0.385} with Qf=(2/3) Cd L √(2g) H1^{3/2}, + O3 mpmath tabulated Q.
Assurance
Engineering
Declared partition coverage
PASS · 5/5 declared partitions (flow, alias, awkward, custom-Cd, invalid-domain) · Matrix
Deferred
Not free-flow thin-plate H only, not V-notch, not ogee Hd, not side weir, not Manning, not Q=Av.
Numerical scope
O2: Villemonte rectangular submerged weir vs a separate-module identity (≤2 ULP). Not free-flow thin-plate H only, not ogee Hd, not side weir, not Manning, and not Q=Av. ≤2 ULP vs O3 applies only to the published tabulated submerged-weir vectors. Not free-flow thin-plate H only, not ogee Hd, not side weir, not Manning, and not Q=Av.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Rectangular submerged weir discharge Qs or downstream head H2 (Villemonte 1947).
Scope
Rectangular thin-plate; drowned nappe
Verification
Engine tested · Source checked · v1.0.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Villemonte 1947 rectangular submerged weir Qs=Qf [1−(H2/H1)^{1.5}]^{0.385} with Qf=(2/3) Cd L √(2g) H1^{3/2}, + 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 94cee226d3df
Legacy regression
6/6 tests · Production surface contract 6/6
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 · 1/1 CVP boundary · 3/3 invalid · 1/1 property · 1/1 metamorphic · 8/8 O3 · 2/2 cross-interface · 6/6 CVP contract · Manifest
Sources
Sources
Evidence
1 legacy golden · legacy regression suite · 4/4 oracle-backed golden · 3/3 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.

Submerged dischargeQs = Qf [1 − (H2/H1)^{1.5}]^{0.385}
Free-flow QfQf = (2/3) C_d L √(2g) H1^{3/2}
iSI g = 9.80665 m/s² unless given. Default C_d = 0.62. Villemonte n=1.5 is the rectangular exponent, not Manning n. H2/H1 below 0.20 is free-flow thin-plate at /calc/physics/weir-discharge.

How to use

1

Enter crest length L

Thin-plate crest in metres. This is not broad-crested b and not side-weir B.

2

Enter upstream head H1

Head above the crest on the upstream side. Do not send free-flow H without H2.

3

Enter downstream head H2 or Qs

Give one. H2 is also above the crest. Invert H2 from Qs on the H2 tab.

4

Optional C_d

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

Example calculations

Common configurations with formula and result.

ϟ

Half submerged

L=1 m, H1=0.3 m, H2=0.15 m, default C_d=0.62

Qs = Qf [1 − 0.5^{1.5}]^{0.385}
Qs≈0.25428 m³/s
ϟ

Invert H2 from that Qs

L=1 m, H1=0.3 m, Qs≈0.25428 m³/s

H2 from Villemonte invert
H2≈0.1500 m

Submerged Weir calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Villemonte 1947 rectangular submerged weir. Qs = Qf [1−(H2/H1)^{1.5}]^{0.385} with free-flow Qf=(2/3) C_d L √(2g) H1^{3/2}. Default C_d=0.62. Envelope H2/H1∈[0.20, 0.95]. Not free-flow thin-plate H only, not ogee Hd, not side weir B+Q1, not Manning n, not Q=Av.
What it calculates
Rectangular submerged weir discharge Qs or downstream head H2 (Villemonte 1947).
Inputs
  • mode?
  • L
  • H1
  • H2 | Q
  • Cd?
  • g?
Outputs
  • Q
  • Qf
  • L
  • H1
  • H2
  • s
  • Cd
  • g
  • n_exp
Formula
Qs = Qf [1 − (H2/H1)^{1.5}]^{0.385}; Qf = (2/3) C_d L √(2g) H1^{3/2}
Assumptions
  • Rectangular thin-plate; drowned nappe
  • SI g=9.80665 unless given
  • Villemonte n=1.5 and exponent 0.385; n is not Manning n
  • Default C_d=0.62; optional C_d in (0, 1]
  • Not free-flow H only, not ogee Hd, not side weir, not Manning, not Q=Av
Units
  • SI; L, H1, H2 in m; Q in m³/s; g in m/s²
Boundary conditions
  • missing L, H1, or H2/Q when required → MISSING_REQUIRED_INPUT
  • H, Q, L, Cd, or g ≤ 0 → VALUE_MUST_BE_POSITIVE
  • H2 ≥ H1 or H2/H1 > 0.95 → VALUE_OUT_OF_RANGE
  • H2/H1 < 0.20, free-flow H, Hd, B/Q1, n, A/v, b, θ → MIXED_INPUT_ENCODING
  • unknown mode → INVALID_MODE
Example
L=1 H1=0.3 H2=0.15 → Qs≈0.25428
Validation cases

4 published on this page · 6/6 tests · Production surface contract 6/6 · View evidence

  • L=1 H1=0.3 H2=0.15 → Qs ≈ 0.25428
  • mode=head L=1 H1=0.3 Q=0.25428 → H2 ≈ 0.1500
  • L=1 H=0.3 H2=0.15 → MIXED_INPUT_ENCODING
  • L=1 H1=0.3 H2=0.15 n=0.013 → MIXED_INPUT_ENCODING
Sources
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Compute rectangular submerged weir discharge from upstream and downstream heads (Villemonte).

Supported and not supported

Supported — Qs from L, H1, H2 · invert H2 from Qs · API physics.fluid.submerged_weir

Not supported — free-flow thin-plate H only; ogee Hd; side weir B+Q1; Manning n; Q=Av; V-notch

Agent / API notes

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

{ "L": 1, "H1": 0.3, "H2": 0.15 }

Invert H2: { "mode": "head", "L": 1, "H1": 0.3, "Q": 0.25428 }. Errors: MISSING_REQUIRED_INPUT, INVALID_NUMBER, VALUE_MUST_BE_POSITIVE, VALUE_OUT_OF_RANGE, MIXED_INPUT_ENCODING, INVALID_MODE.

Calculator URL stays /calc/physics/submerged-weir. 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 free-flow thin-plate weir?

No. Free-flow L+H with no H2 is /calc/physics/weir-discharge. Sending H2 there is MIXED_INPUT_ENCODING. Sending free-flow H here is MIXED_INPUT_ENCODING.

Is this an ogee spillway?

No. USBR high-overflow ogee from L, H, and design head Hd is /calc/physics/ogee-spillway. Sending Hd 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 B or Q1 here is MIXED_INPUT_ENCODING.

Is this Manning?

No. Uniform-flow Q from n, A, R, and S is /calc/physics/manning-formula. Villemonte n=1.5 is not Manning n. Sending n 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 send a V-notch angle?

No. This seed is rectangular. V-notch thin-plate is /calc/physics/weir-discharge. Sending kind=triangular or θ here is MIXED_INPUT_ENCODING.

Where does this run?

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