Froude Number Calculator
Compute the Froude number for open-channel flow or ship motion using velocity, characteristic length or depth, and gravitational acceleration. Classify the regime as subcritical, critical, or supercritical.
Hydraulics & Fluid Mechanics
Froude Number Calculator
Use the Froude Number Calculator to see whether your flow is subcritical, critical, or supercritical, and to check similarity between laboratory models and full-scale rivers, spillways, or ships. This guide walks through the equations, inputs, assumptions, and common pitfalls so your results are physically meaningful, not just a ratio on the screen.
Quick Start
The Froude number compares flow inertia to gravity. In open-channel form, the calculator typically uses \(\mathrm{Fr} = \dfrac{V}{\sqrt{g D}}\), where \(V\) is mean velocity, \(g\) is gravitational acceleration, and \(D\) is hydraulic depth (often close to flow depth in wide channels). Follow these steps to get a reliable result.
- 1 Pick your scenario. Use the open-channel mode for rivers, canals, flumes, and spillways; use the ship / hydraulic model interpretation when you are matching the behaviour of a hull or a physical model to a prototype.
- 2 Enter geometry and flow. For channels, supply either discharge + width + depth or velocity + depth. The calculator will internally compute the mean velocity and hydraulic depth.
- 3 Check your units. Stay consistent: use m/s with m, or ft/s with ft. The calculator converts everything to SI internally, but mixed units are the fastest way to get nonsense.
- 4 Review the Froude number band. As a rule of thumb: \(\mathrm{Fr} < 1\) = subcritical, \(\mathrm{Fr} \approx 1\) = critical, \(\mathrm{Fr} > 1\) = supercritical. The output panel will label the regime for you.
- 5 Use Quick Stats. Derived values such as mean velocity, hydraulic depth, and specific energy help you judge whether the computed Froude number is realistic for your section.
- 6 Run sensitivity checks. Adjust depth, discharge, or width by ±10–20% to see how easily the regime flips from subcritical to supercritical. This mirrors how design changes or uncertainties affect behaviour.
- 7 Document assumptions. The calculator assumes steady, uniform, one-dimensional flow with constant \(g\). Note departures such as rapidly-varied flow, hydraulic jumps, or complex 3D structures.
Tip: If you only know discharge \(Q\), width \(b\), and depth \(y\), let the calculator compute \(V = Q/A\) and \(D = A/T\). For a wide rectangular channel you can often approximate \(D \approx y\).
Warning: Froude numbers calculated right next to a hydraulic jump or control structure may not reflect the upstream or downstream regime. Always tie the input depth and velocity to a clearly defined section.
Choosing Your Method
There are two main ways engineers use a Froude number calculator: evaluating an existing flow section, or designing a model/prototype pair that should behave similarly. The underlying equation is the same, but the inputs and the interpretation differ.
Method A — Open-Channel Section (Field or Design)
Use this when you have a specific cross-section with known discharge and geometry, and you want to classify the flow regime.
- Directly tied to measured quantities on site: depth, width, and discharge.
- Works for rivers, canals, culverts, spillways, and flumes.
- Integrates nicely with Manning or energy calculations in the rest of your design.
- Assumes uniform velocity across the section, which may be rough in natural streams.
- Requires reasonable estimates of geometry (e.g., effective width and depth).
Method B — Model/Prototype Similarity (Ship or Structure)
Use this when you are matching a scale model to a prototype, or checking whether two flows are dynamically similar.
- Essential for spillway, weir, and ship-hull model testing.
- Lets you convert a prototype velocity to a model velocity given a length scale.
- Highlights when Froude similarity should dominate over Reynolds similarity.
- Requires a well-defined characteristic length \(L\) (e.g., depth, hull length, or critical dimension).
- Small models may struggle to match both Froude and Reynolds numbers simultaneously.
In the calculator, you will usually pick Section-based inputs (depth, discharge, width) for open-channel problems, or Velocity + length scale when you are doing similarity or ship-speed checks.
What Moves the Number the Most
Froude number is dominated by the balance between velocity and depth. Understanding the levers that push \(\mathrm{Fr}\) up or down helps you design more stable channels and interpret calculator outputs intelligently.
Higher velocities increase \(\mathrm{Fr}\) linearly. Steep slopes, smooth linings, and contractions all tend to raise \(V\), pushing flows toward or into the supercritical regime.
The term \(\sqrt{g D}\) sits in the denominator. Deeper flow means higher wave celerity and therefore lower Froude numbers for the same velocity. Shallow water with the same discharge produces much larger Froude numbers.
For a given discharge \(Q\), narrowing the channel increases velocity and typically reduces hydraulic depth, both of which increase \(\mathrm{Fr}\). Trapezoidal versus rectangular shapes can shift depth and top width enough to change the regime.
On Earth, \(g \approx 9.81~\text{m/s}^2\) is effectively constant. But the formula reminds you that Froude-based similarity is tied to gravity waves, which is why it governs free-surface model tests.
Rougher linings and obstructions reduce velocity for a given slope and discharge, lowering \(\mathrm{Fr}\). If your calculator result suggests supercritical flow in a very rough, flat channel, re-check your inputs.
Weirs, gates, drops, and transitions can locally force flow toward critical or supercritical conditions. Compute Froude numbers both upstream and downstream of controls, and interpret the location where your inputs apply.
Worked Examples
These examples mirror how you might use the Froude Number Calculator on real projects. You can plug the same values into the tool and compare the outputs, then adjust to explore “what-if” scenarios.
Example 1 — Subcritical Flow in a Rectangular Channel
- Channel type: Rectangular, concrete-lined
- Width \(b\): 3.0 m
- Flow depth \(y\): 1.2 m
- Discharge \(Q\): 6.0 m³/s
- Gravity \(g\): 9.81 m/s²
Cross-sectional area \(A = b y = 3.0 \times 1.2 = 3.6~\text{m}^2\).
Velocity \(V = Q / A = 6.0 / 3.6 \approx 1.67~\text{m/s}\).
For a wide rectangular channel, hydraulic depth \(D = A / T \approx y = 1.2~\text{m}\), since top width \(T \approx b\).
Wave speed term: \(\sqrt{g D} = \sqrt{9.81 \times 1.2} \approx \sqrt{11.77} \approx 3.43~\text{m/s}\).
\[ \mathrm{Fr} = \frac{V}{\sqrt{g D}} \approx \frac{1.67}{3.43} \approx 0.49 \]
\(\mathrm{Fr} \approx 0.49 < 1\), so the flow is clearly subcritical. Surface waves can travel upstream, and the water surface slope will largely follow the energy grade line.
In the calculator, select an open-channel mode, enter \(Q\), \(b\), and \(y\) in SI units, and confirm that the reported Froude number and regime match the hand solution above.
Example 2 — Supercritical Flow on a Steep Chute
- Channel type: Steep spillway chute
- Flow depth \(y\): 0.40 m
- Mean velocity \(V\): 6.0 m/s
- Gravity \(g\): 9.81 m/s²
For a narrow, fully-filled rectangular chute, assume \(D \approx y = 0.40~\text{m}\).
\[ \sqrt{g D} = \sqrt{9.81 \times 0.40} = \sqrt{3.92} \approx 1.98~\text{m/s} \]
\[ \mathrm{Fr} = \frac{V}{\sqrt{g D}} = \frac{6.0}{1.98} \approx 3.03 \] This is well into the supercritical regime.
Highly supercritical flow implies thin, fast, and potentially aerated flow. Downstream, you will typically need a hydraulic jump or energy dissipator to safely return to subcritical conditions.
If you reduce velocity to 4 m/s while keeping depth at 0.40 m, Froude drops to about 2.0, still supercritical but less extreme. Small changes in depth or velocity can significantly shift regime on steep structures.
Common Layouts & Variations
Different hydraulic layouts produce different ranges of Froude number. The table below gives typical patterns and what to expect from the calculator.
| Layout / Use Case | Typical Froude Range | Design Notes |
|---|---|---|
| Low-slope irrigation canal | 0.2–0.8 (subcritical) | Depth is relatively large for the velocity. Flow is tranquil and controlled by downstream water level or structures. Use calculator to verify that curves and transitions stay subcritical. |
| Natural river reach near normal depth | 0.4–1.0 (subcritical to near-critical) | Wide, irregular cross-sections can make velocity and depth estimates uncertain. Use multiple sections and compare Froude along the reach to identify potential hydraulic jumps or controls. |
| Steep flume or spillway chute | 2–5+ (supercritical) | Flow is thin and fast. Upstream is often near-critical; downstream energy dissipation is mandatory. Check Froude at several stations to ensure structures are sized for expected jump conditions. |
| Sudden contraction in a rectangular channel | Fr increases across the contraction | Reduced width raises velocity and usually lowers depth. The calculator can help you test multiple alternative widths and identify whether the transition will push the flow into a supercritical state. |
| Ship hull moving at service speed | 0.2–0.4 (displacement hulls) | Ship Froude number often uses hull length at waterline as \(L\). Higher values correlate with significant wave-making resistance. Scale models should match \(\mathrm{Fr}\) between model and prototype. |
- Verify that the section you are analysing actually matches the layout category you assume.
- Use several cross-sections rather than a single “representative” geometry for long channels.
- Check whether the highest Froude occurs at transitions, contractions, or just downstream of control structures.
- Make sure the depth you input is consistent with the energy grade line you used elsewhere in design.
- For physical models, document the chosen length scale and confirm that model and prototype Froude numbers match.
- Do not assume Froude similarity alone is sufficient when viscous or air–water interaction effects dominate.
Specs, Logistics & Sanity Checks
Froude number itself is dimensionless, but the inputs come from measurements, drawings, and design assumptions. Treat the calculator as a way to formalize those assumptions and check whether they are internally consistent.
Input Quality & Data Sources
- Use flow depths from rating curves, water-surface surveys, or robust hydraulic computations.
- Discharge should come from gaugings, design floods, or verified model outputs, not rough guesses.
- For irregular channels, approximate an equivalent rectangular section only after sketching the actual geometry.
Model / Prototype Scaling
When you are designing a physical model, choose a length scale \(L_m / L_p\) first. Then enforce \(\mathrm{Fr}_m = \mathrm{Fr}_p\) to find the model velocity:
\[ \frac{V_m}{\sqrt{g L_m}} = \frac{V_p}{\sqrt{g L_p}} \quad \Rightarrow \quad V_m = V_p \sqrt{\frac{L_m}{L_p}} \]
The calculator can perform this algebra for you if you enter prototype values and the chosen length scale.
Sanity Checks on Results
- Most stable canals and rivers are comfortably subcritical; if your design produces \(\mathrm{Fr} > 1.3\) in long reaches, revisit the geometry.
- Near-critical flow is often intentionally used at control structures, but should be localised and well understood.
- Very large Froude numbers on steep structures must be paired with proper energy dissipation downstream.
Use the calculator iteratively: adjust width, depth, and slope to keep Froude numbers in a range that matches your design philosophy and regulatory guidance for each facility type.