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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 Engine tested · Source checked · 5/5 tests · Production surface contract 1/1 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
Model
Rectangular submerged weir discharge Qs or downstream head H2 (Villemonte 1947).
Scope
Rectangular thin-plate; drowned nappe
Verification
Engine tested · 5/5 tests · Production surface contract 1/1 · Source checked · v1.0.0
Named expert review
Optional · Not performed
Sources
Sources
Evidence
5 property · Production surface contract 1/1 · Artifact integrity PASS
Production
Embedded snapshot: STALE · Last attested schema matched 1.0.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · Origin ✓ · Live production status PASS (0 stale; 164 CURRENT) @ 2026-09-18T13:56:15.885Z
Semantic contract
PASS

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 · 5/5 tests · Production surface contract 1/1 · 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: Thin-plate weir, Broad-crested weir, Side weir, Ogee spillway, Manning formula .

Q Thin-plate weir › Q Broad-crested weir › Q Side weir › Q Ogee spillway › Q Manning formula ›
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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.