Froude Number Calculator

Compute the Froude number (Fr) — the ratio of inertial to gravitational forces — for open channel flow, ship hydrodynamics, and wave mechanics. Classify flow regime as subcritical, critical, or supercritical, and visualize the flow state on an interactive canvas.

m/s
Use SI units (m/s)
m
Hydraulic depth for open channels; ship waterline length for naval applications.
Use SI units (m)
m/s²
Use SI units (m/s²). Default 9.81
? Subcritical : v=1.5 m/s, L=0.5 m
? Supercritical : v=4.0 m/s, L=0.5 m
⚖️ Critical : v=√(gL) (Fr=1)
? Ship Design : v=8.0 m/s, L=100 m
?️ Open Channel : v=2.0 m/s, L=0.8 m
Privacy first: All calculations are performed locally in your browser. No data is transmitted or stored.

What is the Froude Number?

The Froude number (Fr) is a dimensionless parameter in fluid mechanics that quantifies the ratio of inertial forces to gravitational forces in a flow. Named after the British engineer William Froude (1810–1879), it is a fundamental scaling parameter for free-surface flows, including open channels, rivers, spillways, and ship hulls.

Fr = v / √(g · L)

where v is flow velocity, g is gravitational acceleration, and L is a characteristic length (typically hydraulic depth or waterline length).

The Froude number governs the behavior of free-surface flows:

  • Fr < 1 (Subcritical flow): Gravitational forces dominate. Flow is tranquil, surface disturbances propagate upstream and downstream. Also known as tranquil flow.
  • Fr = 1 (Critical flow): Inertial and gravitational forces are in balance. The specific energy is minimal for a given discharge.
  • Fr > 1 (Supercritical flow): Inertial forces dominate. Flow is rapid, surface disturbances only propagate downstream. Also called rapid flow or shooting flow.

Historical Context & Scientific Significance

William Froude pioneered the use of scaled models for ship resistance testing, establishing the Froude number as a key similarity parameter. His work demonstrated that wave-making resistance scales with the Froude number, enabling naval architects to predict full-scale ship performance from model tests. The Froude number is now a cornerstone of coastal engineering, hydraulic design, and geophysical fluid dynamics.

In open channel hydraulics, the Froude number controls the transition between subcritical and supercritical flow, which is critical for the design of weirs, sluice gates, and stilling basins. The hydraulic jump—a sudden transition from supercritical to subcritical flow—is one of the most dramatic manifestations of Froude number effects.

Why Use an Interactive Froude Number Calculator?

  • Engineering Design: Quickly assess flow regimes for channel design, spillway sizing, and scour protection.
  • Educational Tool: Visualize the relationship between velocity, length, and gravitational forces. Great for hydraulics and fluid mechanics courses.
  • Naval Architecture: Evaluate ship hull performance and wave-making resistance at different speeds.
  • Environmental Flow: Assess river and estuary conditions for ecological and flood management.

Mathematical Derivation & Computation

The Froude number is defined as the square root of the ratio of inertial force to gravitational force:

Fr² = (inertial force) / (gravitational force) = (ρ v² L²) / (ρ g L³) = v² / (g L)

For open channels, the characteristic length L is the hydraulic depth D = A / T (cross-sectional area divided by top width). For ships, L is the waterline length. The Froude number is analogous to the Mach number in compressible flow: both represent the ratio of flow speed to the speed of propagation of disturbances (gravity waves vs. sound waves).

Our calculator solves Fr = v / √(gL) using double-precision arithmetic, ensuring accuracy to at least 12 significant digits. The flow regime is classified using the standard criteria:

  • Subcritical: Fr < 1.00 (tranquil flow)
  • Critical: 0.999 ≤ Fr ≤ 1.001 (within numerical tolerance)
  • Supercritical: Fr > 1.00 (rapid flow)

Step-by-Step Usage Guide

  1. Enter the flow velocity (m/s) — typical values range from 0.1 to 20 m/s for most applications.
  2. Enter the characteristic length (m) — hydraulic depth for channels, waterline length for ships.
  3. Adjust gravitational acceleration (default 9.81 m/s²) — use 9.78 at the equator or 9.83 at the poles for high-precision work.
  4. Click Calculate & Visualize to compute Fr and display the flow regime.
  5. Use the preset examples to explore typical scenarios.
  6. The interactive canvas shows a schematic of the flow, with velocity and length scales annotated.

Reference Table: Froude Number Regimes

The following table is derived from standard hydraulic engineering references (Chow, 1959; Henderson, 1966) and validated against experimental data.

Flow Regime Froude Number Characteristic Behavior Common Applications
Subcritical Fr < 1.00 Tranquil, disturbances travel upstream Rivers, canals, reservoirs
Critical Fr = 1.00 Minimum specific energy, transition point Weir crests, flume throats
Supercritical Fr > 1.00 Rapid, disturbances only downstream Spillways, steep channels, hydraulic jumps
High Supercritical Fr > 2.0 Very rapid, standing waves possible High-head spillways, chute flows
Case Study: Ship Hull Design

A naval architect is designing a 120 m container ship targeting a service speed of 12 m/s (≈23 knots). Using the Froude number with L = 120 m and g = 9.81 m/s²:

Fr = 12 / √(9.81 × 120) = 12 / √1177.2 = 12 / 34.31 = 0.350

This subcritical Fr indicates that the ship operates in a regime where wave-making resistance is moderate. The architect can use this information to optimize the hull form and estimate power requirements. Our calculator provides this instant analysis, allowing rapid iteration of design parameters.

Case Study: Hydraulic Jump in a Stilling Basin

A spillway discharges water at v = 8.0 m/s with a hydraulic depth L = 0.6 m. The Froude number is:

Fr = 8.0 / √(9.81 × 0.6) = 8.0 / √5.886 = 8.0 / 2.426 = 3.30

This supercritical flow (Fr > 1) must be dissipated to prevent downstream erosion. A hydraulic jump is designed to transition the flow to subcritical, with the sequent depth ratio calculated from the Froude number. Our tool quickly identifies the regime, enabling engineers to size the stilling basin effectively.

Common Misconceptions

  • Fr depends only on velocity: False — length scale and gravity are equally important. A small stream at 2 m/s may be supercritical, while a large river at the same speed is subcritical.
  • Critical flow occurs at Fr = 1 only: Correct — but the critical depth depends on discharge and cross-section geometry, not just velocity.
  • Fr is the same as Reynolds number: No — Reynolds number (Re) compares inertial to viscous forces, while Fr compares inertial to gravitational forces. Both are important but govern different phenomena.
  • Froude number is only for open channels: While most common there, it also applies to ships, waves, stratified flows, and even some atmospheric phenomena.

Applications Across Engineering Disciplines

  • Hydraulic Engineering: Design of channels, culverts, weirs, and energy dissipators.
  • Coastal Engineering: Wave run-up, overtopping, and sediment transport.
  • Naval Architecture: Ship resistance, propulsion, and seakeeping.
  • Environmental Fluid Mechanics: Density currents, river mixing, and estuary circulation.
  • Aerospace: Free-surface flows in fuel tanks and water-landing vehicles.

Rooted in classical fluid mechanics – This tool implements the Froude number formulation as established by William Froude and formalized in modern hydraulic engineering (Chow, 1959; Henderson, 1966; French, 1985). The calculation engine and flow visualization have been validated against standard textbooks and experimental data. Reviewed by the GetZenQuery tech team, last updated July 2026.

Frequently Asked Questions

The Froude number represents the ratio of inertial forces (due to flow velocity) to gravitational forces (due to the weight of the fluid). It determines whether surface disturbances can propagate upstream (Fr < 1) or are swept downstream (Fr > 1).

For open channels, use the hydraulic depth D = A / T (cross-sectional area divided by top width). For ships, use the waterline length Lwl. For waves, use the wavelength λ. The choice depends on the specific application and the physical phenomena being studied.

At critical flow, the specific energy (energy per unit weight relative to the bed) is minimized for a given discharge. Small disturbances can cause the flow to switch between subcritical and supercritical regimes. This condition is often used as a control point in hydraulic structures like weirs and flumes.

Yes — the Froude number depends on the characteristic length L. This is why model tests in hydraulics must be designed to match the Froude number between model and prototype to ensure dynamic similarity for gravity-driven flows.

The Froude number is most relevant for incompressible, free-surface flows. For compressible flows, the Mach number is more appropriate. For non-Newtonian fluids, additional rheological parameters are required. This calculator assumes Newtonian, incompressible flow with a free surface.

Key references include: Chow, V.T. (1959) Open-Channel Hydraulics; Henderson, F.M. (1966) Open Channel Flow; and French, R.H. (1985) Open-Channel Hydraulics. Online resources include the USGS Water Science School and the IAHR (International Association for Hydro-Environment Engineering and Research).
References: Britannica: Froude number; Chow, V.T. "Open-Channel Hydraulics" (1959); Wikipedia: Froude number; Henderson, F.M. "Open Channel Flow" (1966).