A weir is an obstruction placed across an open channel that raises the water level so the flow has to spill over its crest. From the depth of water above the crest (the head) the discharge can be inferred. A rectangular weir has a simple horizontal crest and is good for a wide flow range. A V-notch (Thomson) weir has a triangular notch and is well suited to accurate measurement of small flows. Both are standard equipment in labs, irrigation canals and water treatment plants.

How do you choose between a rectangular and a V-notch weir?

Choose by flow range. A rectangular weir works well for medium-to-large flows (from a few L/s up to several m^3/s) and can be made wider for larger discharges. A V-notch weir, where Q scales with h to the 5/2 power, keeps the head above the crest large enough at low flows that read-out errors do not dominate. In recirculating laboratory flumes the 90-degree V-notch is the de facto standard; in main irrigation canals a rectangular weir is more common.

Why does a V-notch weir's discharge scale with h to the 5/2 power?

Because the wetted area of a triangular notch scales with h squared and the velocity of overflow scales with the square root of h. Combining Q = velocity x area gives Q proportional to h^(1/2) x h^2 = h^(5/2). For a rectangular weir the width b of the notch is fixed, so the area scales linearly with h, giving Q proportional to h^(3/2). The larger exponent means a V-notch is far more sensitive to small changes in flow, which is why it dominates low-flow measurement.

Why does the Rehbock discharge coefficient depend on the weir height P?

The Rehbock correction C_d = 0.602 + 0.083 (h/P) accounts for the approach velocity upstream of the weir. When the weir is short (P small) or the head h is large, the water has noticeable speed in the upstream channel and that kinetic energy adds to the overflow. The correction term increases C_d to capture this. It is generally valid up to about h/P = 1; beyond that, separate correlations or physical-model tests are needed.

Weir Flow Simulator — Free Online Calculator

Parameters

Head over crest h

Rectangular crest width b

Upstream weir height P

V-notch angle θ

The Rehbock coefficient C_d varies with h/P; the Thomson formula uses C_d ≈ 0.585 (90° notch). g = 9.81 m/s².

Results

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Rectangular weir discharge Q_rect

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V-notch weir discharge Q_tri

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Rectangular discharge coefficient C_d

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Q_rect / Q_tri

Rectangular and V-Notch Weir Cross Sections

Left = rectangular weir cross section (head h, weir height P, width b) / Right = V-notch cross section (notch angle θ, head h) / Cyan arrow = overflow

Theory & Key Formulas

A weir is an obstruction placed across an open channel that raises the upstream water level so the flow spills over its crest. The discharge can be inferred from the depth (head) of water above the crest.

Rectangular weir (Rehbock formula). b is the crest width, h the head, g the gravity, C_d the discharge coefficient:

$$Q_\text{rect} = C_d\,b\,\sqrt{2g}\,\frac{2}{3}\,h^{3/2}$$

Rehbock correction for the discharge coefficient (P is the upstream weir height):

$$C_d \approx 0.602 + 0.083\,\frac{h}{P}$$

V-notch (Thomson) weir, where θ is the full opening angle:

$$Q_\text{tri} = C_d\,\frac{8}{15}\,\sqrt{2g}\,\tan\!\left(\frac{\theta}{2}\right)\,h^{5/2}$$

For the rectangular weir Q is proportional to h^(3/2); for the V-notch it is proportional to h^(5/2). The larger exponent makes the V-notch much more sensitive at low flows.

What is the Weir Flow Simulator

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I sometimes see a vertical plate sticking up across a small channel with people measuring the water level over it. What is that for?

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That is a "weir", a flow-measuring device. The trick is to put a plate (the weir plate) across the channel on purpose, dam the water a little, and then back-calculate the discharge from the depth of water flowing over the top — the head h. Directly measuring "how many litres per second" is hard, but measuring depth only needs a ruler. Move the "Head over crest h" slider in the simulator above and you'll see the discharge cards swing significantly.

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Which is the "better" weir, rectangular or V-notch?

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It depends on what you want. For large flows, use a rectangular weir; for accurate measurement of small flows, use a V-notch. The reason lives in the discharge-head relation: rectangular gives $Q \propto h^{3/2}$ while V-notch gives $Q \propto h^{5/2}$. The larger exponent of the V-notch means that even a small change in flow produces a clear change in head h. The 90-degree V-notch is essentially the standard for measuring pump discharge in laboratory flumes.

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When I move the "crest width b" slider, only the rectangular discharge changes. The V-notch doesn't react?

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Exactly. V-notch discharge depends only on θ (the notch angle) and h, so the width b is irrelevant. Conversely, moving the θ slider changes only the V-notch discharge while leaving the rectangular one untouched. Working through which formula depends on which parameter while wiggling the sliders is the fastest way to internalize them.

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What is the "upstream weir height P" that shows up in the Rehbock formula?

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It is the height from the upstream channel bottom to the top of the weir plate. When P is small, the water flows noticeably fast in the upstream channel, and that approach velocity adds a little extra to the overflow. The correction term $C_d \approx 0.602 + 0.083 \cdot (h/P)$ in the Rehbock formula captures this. Reduce P in the simulator and the C_d card rises; Q_rect follows. At h/P = 0.5 you get C_d ≈ 0.635.

Frequently Asked Questions

Measure the difference between the water surface well upstream of the weir (about 3 to 4 h or more upstream) and the crest level. Right next to the weir the surface has already started to draw down, so measuring there underestimates h and the discharge. In practice a "stilling well" is installed upstream of the weir and a gauge or sensor reads the water level inside the well.

If the downstream water level rises above the crest, the weir becomes "drowned" (submerged) and neither the Rehbock nor the Thomson formula applies. The discharge coefficient drops because of downstream backwater and a separate correction is required. As a flow-measuring device the accuracy suffers a lot, so designers keep a sufficient drop downstream to maintain free overflow. The simulator assumes free overflow.

Yes. 90° is the standard, but for very small flows 30° and 45° notches are also used. The smaller the angle, the larger h becomes for the same flow, which improves the head-reading accuracy. Wider notches like 120° are an intermediate between rectangular and triangular weirs. The simulator lets θ vary from 10° to 120°.

Manning's equation estimates the discharge of "steady uniform flow" in a long channel with a constant depth. The weir formula instead computes the instantaneous discharge as water spills over a local obstruction and is derived from energy conservation (Bernoulli's equation). Manning suffers from uncertainty in the roughness coefficient and is rarely accurate in absolute terms, whereas a properly installed weir gives discharge to within a few percent once its geometry is fixed. Use a weir to measure flow, and uniform-flow calculations to study channel hydraulics.

Real-World Applications

Laboratory recirculating flumes: A 90-degree V-notch weir at the outlet of a return tank is the workhorse of university and research hydraulics labs. With the notch angle and water depth alone, the discharge can be pinned down to about 1 percent, making the V-notch indispensable for calibrating recirculating pump flows before and after model tests.

Agricultural and irrigation canals: In irrigation systems that branch from a main canal down to fields, rectangular weirs (and Parshall flumes) are placed at each split to manage the water apportioned to each user. They give the documentary evidence needed to share water fairly across the served area, which is why they are basic infrastructure in agricultural engineering.

Sewage and water treatment plants: Rectangular and V-notch weirs are used to gauge the inflow and outflow of each treatment tank. They often coexist with electronic flow meters (electromagnetic or ultrasonic), where the weir provides a calibration reference and a redundant backup measurement.

River observation and hydrology: For continuous discharge measurement of small mountain streams, permanent concrete observation weirs are installed. A water-level gauge upstream of the weir converts level data into discharge in real time, building up the long-term runoff records and flood peaks that underpin river-management and water-resource planning.

Common Misconceptions and Cautions

The most common misconception is to assume that "doubling the head h doubles the discharge". For a rectangular weir $Q \propto h^{3/2}$, so doubling h multiplies the discharge by 2^1.5 ≈ 2.83. For a V-notch $Q \propto h^{5/2}$, so doubling h gives 2^2.5 ≈ 5.66 times the flow. Move h from 0.20 m to 0.40 m in the simulator and Q_tri leaps by about 5.7 times. The intuition that depth and flow are linear simply does not hold in weir hydraulics.

The next most common error is to think that the head can be read anywhere upstream. In fact, within a few h of the weir the water surface is already drawing down toward the crest; measuring there underestimates h. And if there isn't enough drop downstream the flow becomes submerged, and neither the Rehbock nor the Thomson formula applies. A weir works as a flow gauge only when the upstream straightening reach, the downstream free fall, and the correct measurement position are all in place.

Finally, note that the coefficients used by this simulator are idealized representative values. The Rehbock formula $C_d = 0.602 + 0.083\,(h/P)$ is a representative approximation for sharp-crested rectangular weirs up to h/P ≈ 1; real coefficients shift by a few percent with crest thickness, the state of upstream flow, and the aeration of the nappe. Thomson's $C_d \approx 0.585$ is also a representative value for a 90° notch and varies slightly with notch angle and Reynolds number (viscous effects at very low flows). In the field, every installed weir is calibrated against an independent reference flow.

Weir Flow Simulator — Rectangular and V-Notch

What is the Weir Flow Simulator

Frequently Asked Questions

Real-World Applications

Common Misconceptions and Cautions

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