Compute the Froude number from flow velocity and characteristic length, classify open-channel flow as subcritical, critical, or supercritical, or solve for a missing variable — Fr = v/√(gL).
Flow properties
For open-channel flow, use the hydraulic depth. For a ship hull, use the waterline length.
Defaults to standard Earth gravity, 9.81 m/s².
Froude number
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Enter flow velocity and characteristic length to compute.
Flow regime—
Distance to critical (Fr = 1)—
Classify by Froude number
Paste an Fr value from the first tab, or type one directly.
Fr below 1 is subcritical (deep, slow, gravity-dominated flow), Fr equal to 1 is critical (a hydraulic jump forms at this transition), and Fr above 1 is supercritical (shallow, fast, inertia-dominated flow).
Flow regime
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Enter a Froude number to classify the flow.
Regime thresholds
Regime
Fr range
Subcritical
Fr < 1
Critical
Fr = 1
Supercritical
Fr > 1
The critical Froude number marks the point where a hydraulic jump forms as flow transitions between regimes.
Solve for a missing variable
Solved value
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Choose a variable, set the target Fr, and enter the other two values.
4 min read3 steps7 terms3 examples6 FAQsFr = v / √(gL)
The Froude number is the open-channel-flow counterpart to the Reynolds number — instead of comparing inertial forces to viscous friction, it compares them to gravity.
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Walk-through
How to Use This Calculator
3 steps▸
1
Enter flow velocity and length
On the Froude Number tab, enter the flow velocity (m/s) and the characteristic length (m) — hydraulic depth for open-channel flow, or waterline length for a ship hull. Gravitational acceleration defaults to standard Earth gravity, 9.81 m/s², but you can override it.
2
Read the Froude number and regime
The result card shows Fr along with the flow regime — subcritical, critical, or supercritical — and how far the value sits from the critical threshold at Fr = 1. The Flow Regime tab lets you classify any Fr value directly, with a gauge showing where it falls relative to the critical marker.
3
Solve for a missing variable
On the Solve tab, pick the variable you don't know (velocity, characteristic length, or gravity), set a target Froude number, and fill in the other two — the calculator rearranges Fr = v/√(gL) to find the missing value.
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Reference
Formula & Methodology
2 formulas▸
Froude number
Fr = v / √(gL)
Flow velocity v (m/s) divided by the square root of gravitational acceleration g (m/s²) times a characteristic length L (m, typically hydraulic depth or waterline length). Fr is dimensionless — it carries no units.
Subcritical flow is deep, slow, and gravity-dominated — small disturbances can travel upstream. Supercritical flow is shallow, fast, and inertia-dominated — disturbances only travel downstream. The critical point (Fr = 1) marks the boundary, where a hydraulic jump forms as flow transitions between the two.
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Glossary
Key Terms Explained
7 terms▸
Froude number ↗A dimensionless quantity that compares inertial forces to gravitational forces in a flow, used to classify open-channel flow and predict wave-making resistance for ships and hulls.
Open-channel flow ↗Flow with a free surface exposed to the atmosphere, such as a river, canal, or spillway, as opposed to pressurized pipe flow. The Froude number is the primary tool for classifying this type of flow.
Subcritical ↗Flow with Froude number less than 1 — relatively deep and slow, where gravitational forces dominate. Surface disturbances can propagate both upstream and downstream.
Supercritical ↗Flow with Froude number greater than 1 — relatively shallow and fast, where inertial forces dominate. Surface disturbances can only propagate downstream, never upstream.
Critical flow ↗Flow at exactly Fr = 1, the boundary between subcritical and supercritical regimes. At this point, flow velocity equals the speed of a small gravity wave on the surface.
Hydraulic jump ↗An abrupt, turbulent rise in water surface that occurs when flow transitions from supercritical to subcritical, dissipating energy. Hydraulic jumps form at spillways, stilling basins, and downstream of sluice gates.
Hull speed ↗The approximate speed at which a displacement-hull vessel's bow and stern wave crests align, historically treated as a practical speed limit for that hull. It corresponds to a Froude number of roughly 0.4 based on waterline length.
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Scenarios
Real-World Examples
3 worked examples▸
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Open-channel flow
A stream 0.8 m deep flowing at 2 m/s
Velocity 2 m/sLength 0.8 mGravity 9.81 m/s²
Fr = 2 / √(9.81 × 0.8) ≈ 0.714. That's below 1, so the flow is subcritical — relatively deep and slow, with surface waves able to travel upstream against the current.
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Ship hull speed
A 25 m waterline-length displacement hull near its hull speed
Velocity 6.26 m/sLength 25 mGravity 9.81 m/s²
Fr = 6.26 / √(9.81 × 25) ≈ 0.4 — the classic hull-speed Froude number for displacement vessels, where bow and stern wave crests align and wave-making resistance rises sharply beyond this point.
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Hydraulic jump
Spillway flow tuned to exactly the critical threshold
Velocity 3.13 m/sLength 1 mGravity 9.81 m/s²
Fr = 3.13 / √(9.81 × 1) ≈ 1.0 — exactly critical. In practice, spillway and channel flow crossing this threshold from supercritical to subcritical forms a visible hydraulic jump, a turbulent standing wave that dissipates energy.
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Reference
Cite This Calculator
APA & MLA▸
Use either format to cite this calculator in a paper, report, or resource list.
The Froude number is the open-channel-flow counterpart to the Reynolds number — instead of comparing inertial forces to viscous friction, it compares them to gravity. That single ratio predicts whether a river, canal, or ship hull is moving in a deep, gravity-dominated regime or a shallow, inertia-dominated one, and it pinpoints exactly where a hydraulic jump will form. This calculator applies Fr = v/√(gL) to your own flow conditions, classifies the result, and can solve backward for a missing variable when you know the Froude number you're targeting.
How the Froude Number Calculator works
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The Froude number is a ratio of a flow's velocity to the speed of a small gravity wave traveling on its surface, √(gL). When Fr is below 1, the flow moves slower than that wave speed, so surface disturbances can propagate both upstream and downstream — this is subcritical flow. When Fr exceeds 1, the flow outruns its own surface waves, so disturbances can only travel downstream — this is supercritical flow. The transition at Fr = 1 is not gradual: it shows up physically as a hydraulic jump, an abrupt, turbulent rise in surface elevation that dissipates the flow's excess kinetic energy.
Inputs and what they mean
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Velocity (m/s) is the average flow speed. Characteristic length (m) depends on context: for open-channel flow it's usually the hydraulic depth (cross-sectional flow area divided by the free-surface width), while for a ship hull it's the waterline length. Gravitational acceleration (m/s²) defaults to standard Earth gravity, 9.81 m/s², and rarely needs adjustment — but the field exists for specialized modeling contexts. Because L sits under a square root, doubling it only raises the denominator by about 41%, while doubling velocity doubles Fr directly — velocity is usually the input with the biggest leverage on the result.
Limits and edge cases
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The Froude number assumes a single, well-defined free surface and a length scale that meaningfully represents the flow's depth or hull geometry — it doesn't capture three-dimensional effects, wave interference between multiple hulls, or compressibility. For ships, the Fr ≈ 0.4 hull-speed rule of thumb is a useful planing/displacement guideline, not a hard physical limit — many hulls operate somewhat above or below it depending on hull form. For channel and spillway design, always cross-check hydraulic-jump predictions against site-specific hydraulic modeling; this calculator gives the classification and threshold, not a full energy-loss or jump-length calculation.
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Questions
Frequently Asked Questions
6 questions▸
What is the formula for the Froude number?+
Fr = v/√(gL), where v is flow velocity (m/s), g is gravitational acceleration (m/s², typically 9.81), and L is a characteristic length such as hydraulic depth or waterline length (m). Fr is dimensionless.
What does subcritical flow mean?+
Subcritical flow has a Froude number less than 1. It's relatively deep and slow, dominated by gravity, and surface disturbances can travel both upstream and downstream — this is typical of most rivers and canals under normal conditions.
What does supercritical flow mean?+
Supercritical flow has a Froude number greater than 1. It's relatively shallow and fast, dominated by inertia, and surface disturbances can only travel downstream — common just below spillway crests, sluice gates, and steep chutes.
What happens at a hydraulic jump?+
A hydraulic jump occurs where flow crosses the critical threshold, Fr = 1, transitioning from supercritical to subcritical. The abrupt rise in surface elevation dissipates kinetic energy as turbulence, which is why engineers deliberately design stilling basins to trigger jumps and protect downstream channels from erosion.
How does the Froude number apply to ships?+
For displacement-hull vessels, the Froude number based on waterline length predicts wave-making resistance. Around Fr ≈ 0.4, the bow and stern wave systems align in a way that historically defined a practical 'hull speed' — pushing much faster requires disproportionately more power because wave-making drag rises steeply.
Is the Froude number dimensionless?+
Yes. Every unit in v/√(gL) cancels out (m/s divided by √(m/s² × m) reduces to a pure number), which is why it can be used to compare flows of completely different scales — a lab model and a full-size river — on the same subcritical-to-supercritical scale.
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