What is the equation for flow over a weir?

For a rectangular crest, discharge varies with head to the power 1.5: Q = C L H^1.5, where L is the crest length and H the head above the crest. For a sharp-crested weir C is written as Cd × (2/3) × √(2g), with Cd about 0.61 to 0.65. A V-notch varies with head to the power 2.5: Q = Cd × (8/15) × √(2g) × tan(θ/2) × H^2.5.

What discharge coefficient should I use for a broad-crested weir?

In the SI form Q = C L H^1.5, the tool's broad-crested coefficient runs from 1.49 when the head is small compared with the crest breadth to 1.85 when the head is three times the breadth. In US customary units the same range is 2.70 to 3.35. Calibrated field values beat any table, so use them when a structure has been rated.

How do you calculate flow through a 90 degree V-notch weir?

Use Q = Cd × (8/15) × √(2g) × tan(θ/2) × H^2.5 with tan(45°) = 1. With the Kindsvater-Shen coefficient of 0.578 that the tool applies to a 90° notch, a head of 0.20 m gives 0.578 × 0.533 × 4.43 × 0.0179 = 0.0244 m³/s, or 24.4 litres per second. Head is measured above the notch vertex.

What is a Cipolletti weir?

A Cipolletti weir is a sharp-crested trapezoidal weir with side slopes of 1 horizontal to 4 vertical. The sloping sides add about as much flow as the end contractions of a rectangular notch take away, so a single fixed coefficient applies with no contraction correction: Q = 1.859 L H^1.5 in SI units, or 3.367 L H^1.5 in US customary units.

How does submergence affect flow over a weir?

Once tailwater rises above the crest, the weir passes less flow for the same upstream head. The Villemonte equation scales the free-flow discharge by [1 − (H2/H1)^n]^0.385, with n = 1.5 for rectangular crests and 2.5 for V-notches. At a submergence ratio of 0.6 a rectangular weir passes about 79% of its free flow; at 0.9, about 48%.

Weir Flow Calculator: Broad-Crested, V-Notch, Sharp-Crested

Weir results
Discharge, Q	3m³/s
Head above crest, H	0.629m
Coefficient, CBroad-crested table (USBR), at H/Lw = 0.42	1.502m½/s
Unit discharge, q = Q / L	0.75m³/s per m
Upstream depth above bed, P + H	1.429m

Using the calculator above

The calculator opens on the broad-crested worked example further down this page, solving for the head that passes 3.0 m³/s, so you can check its numbers against the text. Picking another weir type loads that type’s check from the same section.

Solve for. Discharge from a measured head, or the head a design flow needs.
Crest length L and crest breadth Lw. L is measured across the channel and Lw along the flow (broad-crested only).
Crest height P and channel width B. The height of the crest, or of the V-notch vertex, above the approach bed, and the width of the approach channel. They set the approach velocity and the sharp-crested coefficient.
End contractions. How many crest ends are set in from the channel sides; none means the plate spans the channel.
Design head Hd. Ogee only: the head the crest profile was drawn for.
Tailwater above the crest. Optional. Leave it blank for free flow; a value applies the Villemonte reduction.
Approach velocity. Off by default. Include it to iterate on the energy head.

Each result names the coefficient’s source and warns when the inputs leave its range.

Weir equations by type

Every weir in the tool relates discharge to the head above its crest. Rectangular crests follow a 3/2-power law and triangular notches a 5/2-power law:

Q = C_d\,\tfrac{2}{3}\sqrt{2g}\;L_{eff}\,H^{3/2} \qquad Q = C\,L\,H^{3/2} \qquad Q = C_d\,\tfrac{8}{15}\sqrt{2g}\,\tan\!\left(\tfrac{\theta}{2}\right) H^{5/2}

The first is the rectangular sharp-crested (thin-plate) weir, the second the broad-crested, Cipolletti and ogee form, and the third the V-notch.

Symbol	Meaning	Units
$Q$	Discharge	m³/s (ft³/s)
$H$	Head above the crest, or above the notch vertex	m (ft)
$L$	Crest length across the channel	m (ft)
$L_{eff}$	Effective length of a contracted sharp crest, $L - 0.1\,n_c H$ for $n_c$ end contractions	m (ft)
$C_d$	Dimensionless discharge coefficient	–
$C$	Coefficient with $\tfrac{2}{3}\sqrt{2g}$ absorbed; differs between SI and US customary	$\mathrm{m^{1/2}/s}$ ( $\mathrm{ft^{1/2}/s}$ )
$\theta$	V-notch angle	degrees
$P$	Crest height above the approach channel bed	m (ft)

A $C$ value converts from SI to US customary by multiplying by 1.811, the square root of 3.281. Dimensionless $C_d$ values are the same in both systems.

Weir type	Coefficient in the tool	Typical use
Rectangular, sharp-crested	Rehbock: $C_d = 0.611 + 0.075\,H/P$	Flow measurement in flumes and small channels
Rectangular, broad-crested	$C$ from $H/L_w$ , table below	Field gauging structures, embankment crests
Cipolletti (1H:4V sides)	$C$ = 1.859 SI (3.367 US), fixed	Irrigation canals, no contraction correction
V-notch	Kindsvater-Shen $C_d$ by angle, table below	Low flows
Compound	V-notch below, rectangular above, summed	Wide flow range in one structure
Proportional (Sutro)	$C_d$ = 0.61	Linear head–discharge for dosing and treatment works
Ogee spillway	USBR $C_0$ from $P/H_d$ , adjusted for $H/H_d$	Dam and reservoir spillways
Broad-crested with approach velocity	As broad-crested, on the energy head	Narrow approach channels, low crests
Submerged	Any of the five base types, with Villemonte	Drowned crests
Irregular / roadway	Segment by segment along a station–elevation crest, with the FHWA HDS-5 paved or gravel road coefficient (1.38–1.71 SI)	Road embankments and irregular controls

Approach velocity and submergence

Where the approach channel is narrow or the crest is low, the velocity head of the approaching flow adds to the measured head. The tool iterates:

V_a = \frac{Q}{B\,(P + H)}, \qquad H_e = H + \frac{V_a^2}{2g}

with $B$ the approach channel width, until $Q$ computed on $H_e$ stops changing.

When tailwater rises above the crest, the Villemonte (1947) equation reduces the free-flow discharge $Q_f$ :

Q_s = Q_f\left[1 - \left(\frac{H_2}{H_1}\right)^{n}\right]^{0.385}

where $H_1$ and $H_2$ are the upstream and tailwater heads above the crest, and $n$ is the exponent of the free-flow equation.

Submergence $H_2/H_1$	Rectangular crest ( $n$ = 1.5)	V-notch ( $n$ = 2.5)
0.2	0.965	0.993
0.4	0.894	0.960
0.6	0.786	0.882
0.7	0.712	0.816
0.8	0.616	0.721
0.9	0.477	0.569

Discharge coefficient tables

V-notch coefficients the tool applies, read from the Kindsvater-Shen curve for fully contracted notches in Bos (1989), interpolated for other angles. The curve ends at 100°; wider notches take the 100° value:

Notch angle	22.5°	30°	45°	60°	90°	100°	120°
$C_d$	0.592	0.586	0.580	0.576	0.578	0.580	0.580

Broad-crested coefficient against head over crest breadth $L_w$ , the crest dimension in the direction of flow:

$H/L_w$	0.1 to 0.3	0.4	0.5	0.6	0.8	1.0	1.2	1.5	2.0	3.0
$C$ (SI)	1.49	1.50	1.51	1.53	1.58	1.65	1.72	1.77	1.82	1.85
$C$ (US)	2.70	2.72	2.74	2.77	2.86	2.99	3.12	3.21	3.30	3.35

Ogee spillway base coefficient at the design head $H_d$ , against crest height over design head:

$P/H_d$	0	0.5	1.0	1.5	2.0	3.0	5.0
$C_0$ (SI)	1.70	1.90	2.00	2.04	2.07	2.11	2.15
$C_0$ (US)	3.08	3.44	3.62	3.70	3.75	3.82	3.89

Worked example: a broad-crested gauging weir

A broad-crested weir spans a 4.0 m wide rectangular channel. The crest is 4.0 m long, 1.5 m broad in the direction of flow and 0.8 m above the bed. The design flow is 3.0 m³/s.

Free flow, no approach velocity: the head is 0.629 m. $H/L_w$ = 0.42, so $C$ = 1.502, and 1.502 × 4.0 × 0.629^1.5 = 3.0 m³/s.
With approach velocity: $V_a$ = 3.0 / (4.0 × (0.8 + 0.615)) = 0.53 m/s and $V_a^2/2g$ = 0.014 m. The head you would measure upstream is 0.615 m; the energy head over the crest is still 0.629 m.
Drowned by a tailwater 0.40 m above the crest: the upstream head rises to 0.720 m. The submergence ratio is 0.556 and the Villemonte factor 0.814.
Sizing: to pass 3.0 m³/s with no more than 0.50 m of head, the crest must be 5.68 m long ( $C$ = 1.49).

The upstream water level in free flow, 0.8 + 0.629 = 1.43 m above the bed, is the control depth for a backwater profile up the channel in the gradually varied flow calculator.

Two smaller checks with the same tool: a 90° V-notch at 0.20 m head passes 0.0244 m³/s. A 1.0 m sharp-crested rectangular weir with two end contractions, 0.5 m high, at 0.25 m head has $C_d$ = 0.611 + 0.075 × 0.5 = 0.649 and $L_{eff}$ = 0.95 m, and passes 0.227 m³/s. A Cipolletti weir with the same 1.0 m crest at the same head passes 0.232 m³/s.

Inputs and where they come from

Head. Measure it far enough upstream to be clear of the drawdown over the crest, at least four to five times the maximum head.
Crest height $P$ and approach width $B$ . From the drawings. They set the approach velocity and the Rehbock coefficient.
Crest breadth $L_w$ . For broad-crested weirs only; it decides the coefficient.
Design head $H_d$ . For ogee crests, the head the crest shape was drawn for.
Tailwater head. For a submerged weir, the downstream level above the crest, often the normal depth of the downstream channel from the Manning’s equation calculator.

Limitations

Below about 0.06 m of head, surface tension and viscosity make every standard coefficient unreliable.
Sharp-crested coefficients assume an aerated nappe that springs clear of the plate. A nappe clinging to the downstream face passes more flow than the equations predict.
Crest height should be at least twice the head for the standard sharp-crested coefficients; below that, include approach velocity.
Villemonte’s correction becomes uncertain at high submergence. Do not rely on a drowned weir for flow measurement above a ratio of about 0.85.
The ogee coefficient away from the design head follows a simplified adjustment. For a dam spillway, follow the full USBR Design of Small Dams procedure and the dam safety evaluation requirements where they apply.
For a road overtopping at a culvert crossing, use the culvert calculator, which balances the flow through the barrel against the flow over the road.

What a free Studio account adds

A free Delta Hydro Studio account includes the Weir Analysis tool, which runs the same engine with more of it exposed:

The weir types the calculator above leaves out: compound (a V-notch below a rectangular crest), proportional (Sutro), and irregular crests or road embankments entered as station–elevation points.
Two more solve modes: a rating curve over a head range you choose, and the crest length that passes a design flow at a maximum head.
A coefficient override for calibrated structures, and a coefficient lookup listing the formula and typical range for every weir type.
The rating curve as a chart you can download, and as a stage–discharge table you can send straight to the Detention Basin Designer for storage routing (the detention pond calculator page explains that method).
The calculation saved to a project and exported as a Word, Excel or PDF report.

Weir flow calculator