Weir Analysis

Open Weir Analysis

Comprehensive weir-flow analysis supporting 10 weir types and 4 solve modes. Compute discharge, head, crest length, or discharge coefficient for sharp-crested, broad-crested, V-notch, Cipolletti, ogee, compound, Sutro, and irregular roadway weirs — including submergence corrections and approach-velocity iteration.

Overview

A weir is a hydraulic structure placed across an open channel to measure, control, or divert flow. Weirs are among the oldest and most reliable flow-measurement devices in hydraulic engineering: they use a simple geometric relationship — between the head of water above a known crest and the resulting discharge — to infer flow from a single depth measurement.

The Weir Analysis tool supports 10 weir types with 4 solve modes, covering the range of practical applications from laboratory flumes to dam spillways and roadway overtopping. This guide describes the theory, governing equations, coefficient selection, submergence effects, and practical design guidance for each type.

All calculations run instantly in the browser — no server round-trip or credits required. Results can be saved and linked to existing projects for reference.

Units

Equations and coefficients throughout this guide use SI units (metres, m³/s, m/s²). Coefficients quoted in the C·L·H^(3/2) form absorb $\sqrt{2g}$; values valid for US-customary units differ numerically. Always confirm the unit system of a tabulated coefficient before substituting.

Weir Types

Rectangular (Sharp-Crested)

A thin-plate weir with a sharp upstream edge that forces the nappe to spring clear of the crest. The most common type for precise flow measurement in laboratories and small channels. Requires full aeration of the nappe for accurate results.

When to use: Flow measurement in controlled settings, laboratory flumes, and small irrigation channels where high accuracy is required.

$$Q \;=\; C_d \cdot \tfrac{2}{3} \cdot \sqrt{2g} \cdot L_{\mathrm{eff}} \cdot H^{3/2}$$

Rectangular sharp-crested weir

Coefficients: $C_d$ typically 0.60–0.62 (Rehbock formula). The effective length $L_{\mathrm{eff}}$ accounts for end contractions: $L_{\mathrm{eff}} = L - 0.1 \, n \, H$, where $n$ is the number of end contractions (0, 1, or 2).

Rectangular (Broad-Crested)

A weir with a crest wide enough for flow to become parallel to the crest, establishing hydrostatic pressure distribution. The nappe does not spring free — instead, critical depth forms on the crest.

When to use: Field flow-measurement structures, dam-spillway crests, and roadway embankments acting as weirs. More robust than sharp-crested weirs in field conditions.

$$Q \;=\; C \cdot L \cdot H^{3/2}$$

Rectangular broad-crested weir

Coefficients: $C$ typically 1.4–1.7 (SI). Depends on the ratio of crest width to head ($t/H$). Values approach 1.7 for ideal broad-crested conditions ($0.08 < H/t < 0.33$).

Cipolletti

A trapezoidal sharp-crested weir with side slopes of 1H:4V (14.04°). The trapezoidal shape compensates for the flow reduction caused by end contractions, eliminating the need for contraction corrections.

When to use: Irrigation-canal flow measurement where a simple formula without contraction corrections is preferred.

$$Q \;=\; 1.859 \cdot L \cdot H^{3/2}$$

Cipolletti weir

Coefficients: The coefficient 1.859 (SI) is fixed — it already accounts for the trapezoidal geometry compensation.

V-Notch

A triangular-notch weir that provides high sensitivity at low flows because a small change in head produces a proportionally larger change in discharge. Common notch angles are 90°, 60°, 45°, and 22.5°.

When to use: Low-flow measurement, laboratory work, and situations where flow varies over a wide range but accuracy at low flows is critical.

$$Q \;=\; C_d \cdot \tfrac{8}{15} \cdot \sqrt{2g} \cdot \tan\!\left(\tfrac{\theta}{2}\right) \cdot H^{5/2}$$

V-notch weir

Coefficients: $C_d$ ranges from 0.58 to 0.62 depending on notch angle and head. For a 90° V-notch, $C_d \approx 0.593$ (Kindsvater-Shen).

90° V-notch

The 90° V-notch is the most widely used and best-calibrated configuration. When in doubt, select this angle for the most reliable coefficient data.

Compound

A weir combining two or more sections of different geometry — typically a V-notch at the bottom for low-flow sensitivity with a rectangular section above for high-flow capacity. The total discharge is calculated by summing contributions from each active section.

When to use: Streams with highly variable flow where both low-flow accuracy and high-flow capacity are needed in a single structure.

$$Q_{\mathrm{total}} \;=\; Q_{\mathrm{v\text{-}notch}} \;+\; Q_{\mathrm{rectangular}} \quad \text{(when } H > \text{notch height)}$$

Compound weir discharge summation

Coefficients: Each section uses its own discharge coefficient. The rectangular-section coefficient should account for the non-standard approach conditions created by the lower V-notch section.

Proportional (Sutro)

A specially shaped weir where the discharge is linearly proportional to the head above a datum. The opening profile follows a hyperbolic curve that produces a linear Q-H relationship above a base rectangular section.

When to use: Chemical-dosing systems, wastewater treatment plants, and any application where a linear head-discharge relationship simplifies control or monitoring.

$$Q \;=\; C_d \cdot L \cdot a \cdot \sqrt{2g} \cdot \left( H - \tfrac{a}{3} \right)$$

Sutro (proportional) weir

Coefficients: $C_d$ typically 0.60–0.62. The parameter $a$ is the height of the rectangular base section.

Ogee Spillway

A spillway crest shaped to match the underside of the nappe from an ideal sharp-crested weir at the design head. This shape eliminates sub-atmospheric pressures on the crest at design conditions, allowing higher discharge coefficients than other weir types.

When to use: Dam spillways, large reservoir outlets, and any application requiring maximum discharge capacity for a given crest length.

$$Q \;=\; C \cdot L \cdot H_{e}^{3/2}$$

Ogee spillway

Coefficients: $C$ ranges from 1.7 to 2.2 (SI). At design head, $C$ approaches 2.18. Below design head the coefficient decreases; above design head, sub-atmospheric pressures increase it slightly but may cause cavitation.

Cavitation risk

Operating an ogee spillway significantly above design head ($H/H_d > 1.3$) can produce sub-atmospheric pressures on the crest that may lead to cavitation damage. Consult USBR Design of Small Dams for high-head operation guidance.

Broad-Crested + Approach Velocity

A broad-crested weir analysis that explicitly accounts for the velocity of approach in the upstream channel. The approach-velocity head is added to the measured head to obtain the total energy head over the weir.

When to use: Situations where the upstream channel is relatively narrow compared to the weir length, or where significant approach velocity exists ($V_a > 0.3$ m/s).

$$Q \;=\; C \cdot L \cdot H_{e}^{3/2} \qquad \text{where} \quad H_{e} \;=\; H + \frac{V_{a}^{2}}{2g}$$

Broad-crested weir with approach velocity

Coefficients: Same coefficient range as the standard broad-crested weir ($C = 1.4$–$1.7$ SI), but iterative computation of approach velocity adjusts the effective head.

Submerged Weir

A weir operating with tailwater elevation above the crest, reducing the effective head and discharge. The Villemonte (1947) equation corrects the free-flow discharge for the submergence ratio.

When to use: Any weir installation where downstream water levels may rise above the weir crest — common in rivers, tidal areas, and backwater-affected channels.

$$Q_{s} \;=\; Q_{f} \cdot \left[ 1 - \left( \frac{H_2}{H_1} \right)^{n} \right]^{0.385}$$

Villemonte submergence correction

Coefficients: The exponent $n$ matches the free-flow equation exponent (3/2 for rectangular, 5/2 for V-notch). The 0.385 exponent is the Villemonte constant.

Irregular / Roadway

An arbitrary cross-section profile acting as a weir — such as a road embankment, levee crest, or natural channel constriction. The crest profile is defined by station-elevation pairs and discharge is computed by integrating the broad-crested weir equation across the submerged crest width.

When to use: Roadway overtopping analysis, levee-breach estimation, and natural channel controls with irregular geometry.

$$Q \;=\; \sum_i C \cdot \Delta L_i \cdot H_{i}^{3/2}$$

Irregular weir — segmented integration

Coefficients: $C$ typically 1.4–1.7 (SI) for broad-crested conditions. FHWA HDS-5 provides guidance for roadway-embankment coefficients based on embankment shape and paving.

Crest-profile input

Enter station-elevation pairs as tab-, comma-, or space-separated values with one point per line. Data can be pasted directly from survey files or HEC-RAS cross-section exports.

Key Equations

Summary of the governing discharge equations for each weir type:

Weir Type	Equation	Exponent
Rectangular Sharp	$Q = C_d \cdot \tfrac{2}{3} \sqrt{2g} \cdot L_{\mathrm{eff}} \cdot H^{3/2}$	3/2
Rectangular Broad	$Q = C \cdot L \cdot H^{3/2}$	3/2
Cipolletti	$Q = 1.859 \cdot L \cdot H^{3/2}$	3/2
V-Notch	$Q = C_d \cdot \tfrac{8}{15} \sqrt{2g} \cdot \tan(\theta/2) \cdot H^{5/2}$	5/2
Ogee	$Q = C \cdot L \cdot H_e^{3/2}$	3/2
Villemonte (submerged)	$Q_s = Q_f \cdot [1 - (H_2/H_1)^n]^{0.385}$	varies

Coefficient form

All equations use SI units (metres, m³/s). If using the simplified form $Q = C \cdot L \cdot H^{3/2}$, ensure the coefficient $C$ includes the $\tfrac{2}{3} \sqrt{2g}$ factor (numerically about 1.70 for $C_d = 0.577$, hence the characteristic “~1.7” broad-crested values).

Coefficient Selection

The discharge coefficient is the most critical parameter in weir analysis. Several well-established sources provide coefficient formulas and tables.

Rehbock (1929)

The Rehbock formula provides the discharge coefficient for full-width rectangular sharp-crested weirs as a function of head and crest height:

$$C_d \;=\; 0.602 + 0.083 \cdot \frac{H}{P}$$

Rehbock discharge coefficient

where $P$ is the crest height above the channel bed.

Kindsvater-Shen (1957)

Extends the Rehbock approach to contracted rectangular and V-notch weirs by introducing effective-length and effective-head corrections. Widely adopted by ISO 1438 and USBR:

$$L_{\mathrm{eff}} \;=\; L + k_L \qquad H_{\mathrm{eff}} \;=\; H + k_H$$

Kindsvater-Shen effective dimensions

where $k_L$ and $k_H$ are small (typically a few millimetres) empirical corrections that account for surface-tension and viscous effects near the crest.

USBR Water Measurement Manual

Provides comprehensive coefficient tables and design criteria for all standard weir types. The primary reference for broad-crested weir coefficients as a function of crest width and head.

FHWA HDS-5

Provides coefficients for roadway embankments acting as broad-crested weirs, including corrections for embankment shape, paving, and submergence. Essential for roadway overtopping analysis.

Typical coefficient ranges

Weir Type	Coefficient	Typical Range
Rectangular Sharp	$C_d$	0.58 – 0.65
Broad-Crested	$C$	1.4 – 1.7 (SI)
V-Notch (90°)	$C_d$	0.58 – 0.62
Ogee Spillway	$C$	1.7 – 2.2 (SI)
Cipolletti	$C$	1.859 (fixed, SI)

Choosing coefficients

When field-calibration data is available, always prefer calibrated coefficients over textbook values. The built-in coefficient lookup in the tool provides recommended values based on the selected weir type and input geometry — but treat them as a starting point rather than a final answer for critical infrastructure.

Submergence Effects

When tailwater rises above the weir crest, the weir becomes submerged and its discharge capacity is reduced. The Villemonte (1947) equation provides a general correction applicable to all standard weir types:

$$Q_{s} \;=\; Q_{f} \cdot \left[ 1 - \left( \frac{H_2}{H_1} \right)^{n} \right]^{0.385}$$

Villemonte submergence correction

Where:

• $Q_s$ = submerged discharge
• $Q_f$ = free-flow discharge (as if unsubmerged)
• $H_1$ = upstream head above crest
• $H_2$ = downstream (tailwater) head above crest
• $n$ = exponent from the free-flow equation (3/2 for rectangular, 5/2 for V-notch)

The submergence ratio $S_r = H_2 / H_1$ determines the severity of the flow reduction:

• $S_r < 0.3$ — effect is minimal (< 2% reduction).
• $S_r = 0.3$–$0.7$ — moderate submergence; discharge is progressively reduced.
• $S_r > 0.7$ — high submergence; reduction becomes very significant and measurement accuracy deteriorates rapidly.

High-submergence warning

When the submergence ratio exceeds 70% ($H_2/H_1 > 0.7$), the Villemonte equation accuracy decreases significantly. Above 85% submergence, the weir essentially behaves as a submerged orifice and weir equations should not be relied upon for flow measurement. Consider alternative measurement methods or relocating the weir.

Approach Velocity

The approach velocity is the mean velocity of flow in the upstream channel approaching the weir. When this velocity is significant, the kinetic-energy head must be added to the measured static head to obtain the total energy head:

$$V_{a} \;=\; \frac{Q}{B \, (P + H)} \qquad H_{e} \;=\; H + \frac{V_{a}^{2}}{2g}$$

Approach velocity and total energy head

Where:

• $V_a$ = approach velocity (m/s)
• $B$ = upstream channel width (m)
• $P$ = crest height above channel floor (m)
• $H$ = measured head above crest (m)
• $H_e$ = effective (total energy) head (m)

Since $Q$ depends on $H_e$ and $H_e$ depends on $Q$ (through $V_a$), the computation is inherently iterative. The tool performs this iteration automatically, converging to a stable solution within a few cycles.

When approach velocity matters

Approach-velocity corrections become important when the channel-to-weir width ratio $B/L < 3$, or when the crest height $P$ is small relative to the head $H$ ($P/H < 2$). In these conditions, neglecting approach velocity can underestimate discharge by 5–15%.

Rating Curves

A rating curve (stage-discharge relationship) shows how discharge varies with head over the weir. The tool generates rating curves by computing discharge at multiple head increments from zero to a specified maximum.

Rating curves serve several practical purposes:

• Design verification: Confirm the weir can pass the design flood at an acceptable head.
• Field calibration: Compare computed curves against measured flow and head data to validate or adjust coefficients.
• Operational tables: Generate head-discharge lookup tables for field operators.
• Sensitivity analysis: Assess how the discharge responds to changes in head across the full operating range.

Charts can be downloaded as PNG images, and the underlying data can be exported as CSV for use in other software or reporting.

Solve Modes

The tool provides four solve modes to address different analysis scenarios:

Mode	Known	Solves For	Example Use
Solve for $Q$	Head, length, coefficient	Discharge (m³/s)	Flow measurement from measured water level
Solve for $H$	Discharge, length, coefficient	Head over crest (m)	Determine water level upstream of a weir for a design flood
Solve for $L$	Discharge, head, coefficient	Required crest length (m)	Size a new weir to pass a design flow at a maximum allowable head
Solve for $C_d$	Discharge, head, length	Discharge coefficient	Back-calculate coefficient from measured flow and head data

Iterative solutions

Solving for $H$ requires an iterative approach since the relationship between $Q$ and $H$ is nonlinear. The tool uses a bisection method that converges to within 0.0001 m accuracy. Solutions for $Q$, $L$, and $C_d$ are computed directly from the governing equations.

Design Considerations

Key factors to consider when designing or selecting a weir.

Minimum head for accuracy

Standard weir equations lose accuracy at very low heads due to surface-tension and viscous effects. Maintain a minimum head of $H > 0.06$ m (60 mm) for both sharp-crested and broad-crested weirs to ensure reliable results.

Crest height (P/H ratio)

The crest height $P$ must be sufficient relative to the head $H$ to ensure proper flow conditions. A ratio of $P/H > 2$ is recommended for accurate measurement. When $P/H < 2$, approach-velocity effects become significant and the standard coefficient formulas may not apply without correction.

Approach conditions

The upstream channel should provide calm, uniform flow conditions at the weir. Locate the head-measurement point at least 4–5 times the maximum head upstream from the weir crest. Avoid measuring head in the drawdown zone immediately upstream of the crest.

Nappe aeration

For sharp-crested weirs, the nappe (overflow sheet) must spring clear of the downstream face and air must be able to circulate freely beneath the nappe. Insufficient aeration causes clinging flow, which increases the effective head and results in higher discharge than predicted by standard equations.

Nappe ventilation

If the nappe clings to the downstream face of a sharp-crested weir, the standard discharge coefficient is no longer valid. Ensure adequate ventilation by providing air vents or maintaining sufficient tailwater clearance below the nappe.

Tailwater clearance

For free-flow conditions, the tailwater level must remain below the weir crest. For sharp-crested weirs the tailwater should also be low enough to allow full aeration of the nappe. If tailwater rises above the crest, use the submerged-weir (Villemonte) correction. A minimum clearance of 0.05 m below the crest is recommended for sharp-crested weirs to maintain free-flow conditions.

Freeboard

Add freeboard above the computed upstream water surface to account for wave action, approach-velocity surges, and debris accumulation. Typical freeboard is 0.15 to 0.60 m depending on structure size and the consequences of overtopping. For dam spillways, refer to dam-safety guidance for formal freeboard determination.

Related Tools

• Channel Analysis — compute uniform-flow depth and velocity in the channel approaching the weir.
• GVF Profiles — trace the backwater profile upstream of a weir in a long channel reach.
• Flood Routing — route a hydrograph through a reservoir controlled by a weir outlet.
• Culvert Designer — where roadway overtopping interacts with culvert inlet control.

References

• USBR. (2001). Water Measurement Manual (3rd ed., revised). U.S. Bureau of Reclamation, Denver. Comprehensive reference for weir design, coefficients, and installation standards.
• USBR. (1987). Design of Small Dams (3rd ed.). U.S. Bureau of Reclamation, Denver. Primary reference for ogee-spillway profiles, discharge coefficients, and cavitation limits.
• USACE. (1992). Hydraulic Design of Spillways. Engineer Manual EM 1110-2-1603. US Army Corps of Engineers. Detailed design guidance for ogee, chute, and side-channel spillways.
• ISO 1438. (2017). Hydrometry — Open Channel Flow Measurement Using Thin-Plate Weirs. International Organization for Standardization, Geneva. International standard for sharp-crested rectangular and V-notch weir measurement.
• FHWA HDS-5. Normann, J.M., Houghtalen, R.J., and Johnston, W.J. (2012). Hydraulic Design of Highway Culverts (3rd ed.). Federal Highway Administration. Includes roadway-overtopping analysis and broad-crested weir coefficients for embankments.
• Rehbock, T. (1929). Discussion of “Precise weir measurements.” Transactions of the ASCE, 93. Foundational coefficient formula for sharp-crested weirs.
• Kindsvater, C.E. & Carter, R.W. (1957). Discharge characteristics of rectangular thin-plate weirs. Journal of the Hydraulics Division, ASCE, 83(HY6). Effective-length and effective-head correction methodology.
• Villemonte, J.R. (1947). Submerged-weir discharge studies. Engineering News-Record, 139(26), 866–869. Original derivation of the submergence correction.
• Chow, V.T. (1959). Open-Channel Hydraulics. McGraw-Hill, New York. Chapter 14 — Spillways and flow-measurement structures.
• SANRAL. (2013). Drainage Manual (6th ed.). South African National Roads Agency, Pretoria.

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Open Weir Analysis