UNDERSTAND IT. WORK IT OUT.

Learn: Triangular V-notch weir

A triangular V-notch weir estimates discharge from upstream head above its notch vertex. The width of flowing water and jet speed both contribute to the head exponent.

Beginner-friendlyFree · No accountTwo worked examples + separate practice
01

What the formula is saying

Use tan(θ/2) for the notch geometry, H^(5/2) for head, and multiply by (8/15)Cd√(2g). The fractional power comes from adding the discharge of horizontal strips through the opening.

Q = (8/15) Cd tan(θ/2) √(2g) H^(5/2)

Read the symbols in plain language

Cd
Discharge coefficient

Discharge coefficient. Combine the head, geometry, gravity and discharge-coefficient contributions.

ratio / no unit

A dimensionless ratio has no physical unit; 0.01 as a ratio is 1% when the percent option is selected.

θ
Notch angle

Notch angle. The V-notch width varies with the tangent of half its full included angle.

deg

Angles are entered in degrees; multiply by π/180 for trigonometric calculations in radians.

g
Gravity

Gravity. Combine the head, geometry, gravity and discharge-coefficient contributions.

m/s²

Use m/s² as the base unit shown here. H is in m, g is m/s² and Cd is dimensionless. θ is the full included notch angle in degrees; the tangent uses half that angle converted to radians. The result is m³/s.

H
Head

Head. Apply the correct fractional exponent before multiplying by the remaining factors.

m

Metres measure length; 1 m = 1000 mm.

Q
Result to find

Triangular V-notch weir. Combine the head, geometry, gravity and discharge-coefficient contributions.

m³/s

Sort out the units first

H is in m, g is m/s² and Cd is dimensionless. θ is the full included notch angle in degrees; the tangent uses half that angle converted to radians. The result is m³/s.

Assumptions before calculating

Assume a sharp-crested free-flow weir with a ventilated nappe, negligible approach-velocity correction in this simplified form, and a discharge coefficient appropriate to the installation and measurement range.

02

Let’s solve one together

Read the given values, follow each operation, then check what the result means.

Read the supplied values as one complete study case. Find Q and explain the result in the stated output unit.

Cd · Discharge coefficient
0.62
θ · Notch angle
90 deg
g · Gravity
9.81 m/s²
H · Head
0.3 m

  1. Convert half the notch angle to radians

    The V-notch width varies with the tangent of half its full included angle.

    (90) × π ÷ 360 ≈ 0.7853981634 rad

  2. Calculate the notch geometry factor

    The tangent controls how quickly the triangular opening widens with height.

    tan((0.7853981634)) = 1

  3. Evaluate the head power

    Apply the correct fractional exponent before multiplying by the remaining factors.

    (0.3)^2.5 ≈ 0.04929503018 m^(5/2)

  4. Calculate the weir discharge

    Combine the head, geometry, gravity and discharge-coefficient contributions.

    (8 ÷ 15) × (0.62) × (1) × √(2 × (9.81)) × (0.04929503018) ≈ 0.07220097391 m³/s
Answer0.07220097391 m³/s

Does this worked answer make sense?

Doubling H multiplies predicted flow by 2^(5/2), about 5.66 at unchanged geometry and Cd. This strong sensitivity makes accurate head measurement important.

A second worked example — different values

A second case uses different data. Predict which way the answer will change, then calculate it without reusing the first answer.

Cd · Discharge coefficient
0.6
θ · Notch angle
60 deg
g · Gravity
9.81 m/s²
H · Head
0.25 m

  1. Convert half the notch angle to radians

    The V-notch width varies with the tangent of half its full included angle.

    (60) × π ÷ 360 ≈ 0.5235987756 rad

  2. Calculate the notch geometry factor

    The tangent controls how quickly the triangular opening widens with height.

    tan((0.5235987756)) ≈ 0.5773502692

  3. Evaluate the head power

    Apply the correct fractional exponent before multiplying by the remaining factors.

    (0.25)^2.5 = 0.03125 m^(5/2)

  4. Calculate the weir discharge

    Combine the head, geometry, gravity and discharge-coefficient contributions.

    (8 ÷ 15) × (0.6) × (0.5773502692) × √(2 × (9.81)) × (0.03125) ≈ 0.02557342371 m³/s
Answer0.02557342371 m³/s
03

Now try your own values

Change a value or its unit. The same method will show your calculation, step by step.

Discharge coefficient. Combine the head, geometry, gravity and discharge-coefficient contributions.

Notch angle. The V-notch width varies with the tangent of half its full included angle.

Gravity. Combine the head, geometry, gravity and discharge-coefficient contributions.

Head. Apply the correct fractional exponent before multiplying by the remaining factors.

English, Arabic and Persian digits are supported. The steps convert inputs to the formula’s base units.

Results update only when you calculate. The lesson example above stays unchanged.

04

Your turn — check your understanding

Solve this separate case yourself. Use only the values below; the two worked examples use different data. Give the requested result in the selected unit.

Cd · Discharge coefficient
0.61
θ · Notch angle
90 deg
g · Gravity
9.81 m/s²
H · Head
0.2 m

Find: Learn: Triangular V-notch weir

For repeating decimals, use at least four significant figures. Accepted rounding tolerance: 0.05% of the expected value; zero uses an absolute tolerance of 10⁻¹².

A hint, not the answer

Use tan(θ/2) for the notch geometry, H^(5/2) for head, and multiply by (8/15)Cd√(2g). The fractional power comes from adding the discharge of horizontal strips through the opening.

H is in m, g is m/s² and Cd is dimensionless. θ is the full included notch angle in degrees; the tangent uses half that angle converted to radians. The result is m³/s.

Show the full practice solution

Compare the steps with your work; revealing a solution does not mark the lesson complete.

  1. Convert half the notch angle to radians

    The V-notch width varies with the tangent of half its full included angle.

    (90) × π ÷ 360 ≈ 0.7853981634 rad

  2. Calculate the notch geometry factor

    The tangent controls how quickly the triangular opening widens with height.

    tan((0.7853981634)) = 1

  3. Evaluate the head power

    Apply the correct fractional exponent before multiplying by the remaining factors.

    (0.2)^2.5 = 0.01788854382 m^(5/2)

  4. Calculate the weir discharge

    Combine the head, geometry, gravity and discharge-coefficient contributions.

    (8 ÷ 15) × (0.61) × (1) × √(2 × (9.81)) × (0.01788854382) ≈ 0.02577822759 m³/s
Answer0.02577822759 m³/s

Avoid the common trap

Do not measure H at the falling nappe where drawdown occurs; use the prescribed upstream measurement location. Do not use the half-angle as the input and then halve it again, or use exponent 3/2 for a V-notch.

When this method applies — and when it does not

This is a coefficient-based teaching equation, not a substitute for a calibrated installation standard. The angle must lie strictly between 0° and 180°, but practical calibrated angle and head ranges are narrower. Submergence, debris, unventilated flow and very small heads can invalidate the assumptions.

For study and understanding, not approval of a real structure, site operation or design. Apply the correct standard, National Annex and professional review to actual engineering work.

One idea understood. Keep going.

This optional checkmark is saved only in this browser. It is your own progress note, not a certificate.

Sources & further reading

References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.

Reading focus: Triangular V-notch weir. Water-measurement principles; for discharge devices, read the orifice/weir chapters and the installation and head-measurement conditions, not only the coefficient formula.

Lesson updated: · Both examples and the separate practice case are checked against an independent high-precision numerical implementation. This verifies arithmetic for the stated model, not engineering certification.

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