Froude Number Calculator — Open Channel Flow & Ship Hull Analysis
Calculate the Froude number for open channel flow, spillways, and ship hulls. Determine if flow is subcritical, critical, or supercritical with wave celerity.
Froude Number
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The Formula
The Froude number is the ratio of flow inertia to gravitational force. Fr < 1 = subcritical (tranquil) flow where gravity dominates and waves travel upstream. Fr = 1 = critical flow (minimum specific energy). Fr > 1 = supercritical (shooting) flow where inertia dominates and disturbances cannot travel upstream.
Variable Definitions
Flow Velocity
Average cross-sectional velocity. For open channels: v = Q/A where Q is discharge (m³/s) and A is cross-sectional area (m²). For ships: hull speed relative to water.
Gravitational Acceleration
Earth standard: 9.80665 m/s². Differs on other planets. The key insight: Fr ties flow behavior to the gravitational environment — the same flow on Mars has a different Fr.
Characteristic Length
Depends on application. Open channels: hydraulic depth D = A/T (area/top width). Ships: waterline length L_wl. Spillways and weirs: head H above crest. The choice of L affects the numerical value of Fr.
How to Use This Calculator
- 1
Enter the flow velocity in m/s.
- 2
Use the default Earth gravity (9.81) or enter a custom value for other planets.
- 3
Enter the characteristic length — choose based on your application.
- 4
Check the flow regime to determine if the flow is subcritical, critical, or supercritical.
Quick Reference
| From | To |
|---|---|
| Fr < 1 | Subcritical — tranquil, deep, slow |
| Fr = 1 | Critical — min specific energy |
| Fr > 1 | Supercritical — rapid, shallow, shooting |
| Typical river | Fr ≈ 0.1–0.5 (subcritical) |
| Mountain stream | Fr ≈ 0.5–2.0 (mixed) |
| Spillway chute | Fr ≈ 2–6 (supercritical) |
| Ship hull (Fr < 0.4) | Displacement mode |
| Ship hull (Fr > 0.4) | Semi-planing/planing |
Common Applications
- Open channel hydraulics — classifying flow in rivers, canals, and spillways for flood control and irrigation design.
- Naval architecture — ship resistance depends critically on Fr; hull speed is reached at Fr ≈ 0.4 (displacement hulls).
- Hydraulic jump design — stilling basins use hydraulic jumps (Fr > 1 → Fr < 1) to dissipate energy downstream of dams.
- Dam spillway design — ensuring supercritical flow on the spillway face to prevent cavitation and structural vibration.
- Stormwater management — culvert and weir design requires Fr analysis to prevent downstream erosion.
This froude number covers open channel hydraulics. Use the worked examples to verify your understanding and bookmark for quick reference.
Pro Tips
Bookmark this calculator for quick reference — these calculations are frequently needed in engineering workflows.
Verify results against standard handbook values before applying to critical design decisions.
Use the worked examples to confirm your understanding of the underlying formulas.
Understanding the Concept
The Froude number (Fr), named after British engineer William Froude (1810–1879), is a dimensionless number that characterizes the relative importance of inertial forces to gravitational forces in a fluid flow. Froude pioneered the use of scale models for ship design, establishing the law of dynamic similarity: model and prototype behave similarly when Fr is matched. The Froude number governs all free-surface flows — rivers, spillways, ocean waves, and ship wakes. The critical value Fr = 1 represents a flow state of minimum specific energy. At this point, the flow velocity equals the wave propagation speed (celerity), and surface waves become stationary. A hydraulic jump forms when supercritical flow (Fr > 1) transitions to subcritical (Fr < 1), dissipating large amounts of energy. Hydraulic jumps are intentionally created in stilling basins to protect riverbeds downstream of dams. The Froude number also determines ship wave-making resistance: at "hull speed" (Fr ≈ 0.4 for displacement hulls), the wavelength of the bow wave equals the ship length, causing a sharp increase in drag. Planing hulls operate at Fr > 1, riding on top of their own bow wave.
Worked Examples
Water flows at 2 m/s in a rectangular channel 1.5 m deep. Is the flow subcritical or supercritical?
2
9.81
1.5
Result:
Insight: Fr = 2 / √(9.81 × 1.5) = 2 / 3.836 = 0.521. Since Fr < 1, this is subcritical flow. A disturbance (like a rock) would create ripples that travel upstream. The flow is tranquil and controlled by downstream conditions. This is typical for a slow-moving irrigation canal or lowland river.
A dam spillway has water shooting down at 15 m/s with a depth of 0.3 m. What is the Froude number?
15
9.81
0.3
Result:
Insight: Fr = 15 / √(9.81 × 0.3) = 15 / 1.715 = 8.75. This is highly supercritical flow — a "choppy jump" regime. At Fr ≈ 8.75, a hydraulic jump would dissipate about 75% of the incoming energy. A stilling basin is absolutely essential to prevent downstream scour.
Limitations
- This calculator computes the bulk Froude number from average velocity and characteristic length. It does not compute local Fr variations. The characteristic length must match your application — using the wrong L gives a misleading Fr. For non-rectangular channels, use hydraulic depth (A/T) as the characteristic length. Fr does not account for viscous effects; the Reynolds number must be checked separately for complete flow characterization.
Frequently Asked Questions
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