Froude Number Calculator
Enter the flow velocity and hydraulic depth (or channel area and width) to get the Froude number instantly. The calculator classifies the flow regime, solves for critical depth and critical velocity in a rectangular channel, and computes the hydraulic jump sequent depth and energy dissipation if the flow is supercritical. Switch between metric and imperial units at any time.
Formula
Worked example
A rectangular channel carries flow at V = 3 m/s with hydraulic depth h = 1.2 m. Wave celerity c = sqrt(9.806 * 1.2) = 3.431 m/s. Fr = 3 / 3.431 = 0.874, which is subcritical. Specific energy E = 1.2 + 3^2 / (2 * 9.806) = 1.659 m. For unit discharge q = 2.5 m^2/s, critical depth hc = (2.5^2 / 9.806)^(1/3) = 0.860 m.
What is the Froude Number?
The Froude number (Fr) is a dimensionless ratio that compares the inertial force of a flowing liquid to the restoring force of gravity. Named after the nineteenth-century British engineer and naval architect William Froude, it is the most fundamental parameter in open channel hydraulics. The formula is Fr = V / sqrt(g * h), where V is the mean cross-sectional flow velocity, g is gravitational acceleration, and h is the hydraulic depth (cross-sectional area divided by the water-surface width). When Fr < 1 the flow is subcritical (also called tranquil or streaming): gravity dominates and surface disturbances can travel upstream. When Fr > 1 the flow is supercritical (rapid or shooting): inertia dominates and disturbances are carried downstream only. At Fr = 1 the flow is critical and wave celerity exactly equals the flow velocity, which creates an inherently unstable condition.
How to use this calculator
Select the unit system (metric or imperial) and choose what you want to solve: Froude number from velocity and depth, velocity from a target Fr and depth, or hydraulic depth from a target Fr and velocity. Enter the known values and read the outputs. The hydraulic depth for a wide rectangular channel is just the water depth; for trapezoidal or irregular sections use h = cross-sectional area / top width. If you enter a unit discharge (discharge per unit channel width), the calculator also returns critical depth and critical velocity for a rectangular section. For supercritical flows (Fr > 1) the hydraulic jump sequent depth and energy dissipation are computed automatically using the Belanger equation.
Hydraulic Jump, Critical Depth, and Specific Energy
A hydraulic jump is the abrupt transition from supercritical to subcritical flow, always accompanied by turbulent energy dissipation. The sequent (conjugate) depth h2 downstream of the jump is related to the upstream depth h1 and Froude number Fr1 by the Belanger equation: h2 / h1 = 0.5 * (sqrt(1 + 8 * Fr1^2) - 1). The energy lost in the jump rises steeply with Fr1: about 17% at Fr = 2, 56% at Fr = 5. Stilling basins in dam spillways are sized using this relationship to prevent erosion downstream. Critical depth hc is the depth at which specific energy (E = h + V^2 / (2g)) is minimised for a given discharge in a rectangular channel: hc = (q^2 / g)^(1/3), where q is discharge per unit width. At critical depth the Froude number equals exactly 1 and the velocity equals the wave celerity.
Applications in Engineering and Science
Open channel hydraulics: the Froude number governs spillway and weir design, culvert control regime identification, river reach classification, sediment transport intensity, and backwater effect determination. Naval architecture: for ships the characteristic length L is the waterline length and the Froude number compares hull speed to wave speed. Wave-making resistance increases sharply above Fr = 0.4 for displacement hulls, which sets a practical hull-speed limit. Scale model testing uses Froude similarity (model Fr = prototype Fr) to ensure gravity-wave behaviour is reproduced; velocity must scale with the square root of the length ratio. Fluid mechanics broadly: the Froude number plays a role analogous to the Mach number in compressible gas dynamics, with the critical condition (Fr = 1) corresponding to sonic flow.
Froude Number Flow Regime Classification
| Froude Number Range | Regime | Characteristics | Typical Settings |
|---|---|---|---|
| Fr < 0.5 | Deep subcritical | Very tranquil; gravity strongly dominant | Large rivers, estuaries |
| 0.5 - 0.85 | Subcritical | Tranquil; disturbances travel upstream | Normal rivers and channels |
| 0.85 - 1.0 | Near-critical (sub) | Transition zone; unstable | Avoid in design |
| Fr = 1 | Critical | Wave celerity equals flow velocity | Weirs, flumes, control sections |
| 1.0 - 1.15 | Near-critical (super) | Transition zone; unstable | Avoid in design |
| 1.15 - 2.5 | Supercritical | Rapid; inertia dominant, disturbances swept downstream | Spillway aprons, chutes |
| Fr > 2.5 | Strong supercritical | High energy dissipation in hydraulic jump | Stilling basins required |
Standard hydraulic classification of open channel flow by Froude number. The critical zone (Fr near 1) is avoided in practice because it is inherently unstable.
Frequently asked questions
What does a Froude number of 1 mean?
Fr = 1 is the critical condition where the flow velocity equals the speed of shallow-water gravity waves (the wave celerity). At this point, a small disturbance on the water surface remains stationary relative to the channel. Critical flow is the boundary between subcritical and supercritical regimes and is hydraulically unstable: any small perturbation pushes the flow one way or the other. In practice it is deliberately created only at control structures such as weirs, Parshall flumes, and critical-flow flumes.
What is hydraulic depth and how is it different from water depth?
Hydraulic depth h is the cross-sectional flow area A divided by the top (water-surface) width T: h = A/T. For a wide rectangular channel where the width is much larger than the depth, the two are essentially equal. For triangular, trapezoidal, or circular sections they differ. Hydraulic depth is the correct length scale for the Froude number in non-rectangular channels because it accounts for the channel geometry.
How do I find critical depth for a rectangular channel?
For a rectangular channel, critical depth hc = (q^2 / g)^(1/3), where q is the discharge per unit width (total discharge divided by channel width) and g is gravitational acceleration. At critical depth, the Froude number is exactly 1 and the specific energy is at its minimum for that discharge. This calculator computes critical depth automatically when you enter the unit discharge.
What is the Belanger equation used for?
The Belanger equation gives the ratio of depths across a hydraulic jump in a rectangular horizontal channel: h2 / h1 = 0.5 * (sqrt(1 + 8 * Fr1^2) - 1). Here h1 and Fr1 are the upstream depth and Froude number (supercritical side), and h2 is the deeper, slower downstream depth after the jump. It is fundamental in stilling basin design for dam spillways, because the jump converts kinetic energy into heat and turbulence, protecting the channel bed from erosion.
Why is the Froude number important in ship design?
For displacement ships the Froude number replaces hydraulic depth with the waterline length L: Fr = V / sqrt(g * L). Wave-making resistance increases steeply above Fr = 0.4, so conventional displacement hulls are designed to cruise below that. At around Fr = 0.5 a displacement hull is essentially at its hull speed and consumes enormous power to go faster. Planning hulls and hydrofoils bypass this limit by rising above the wave system.
What is wave celerity and how does it relate to the Froude number?
Wave celerity c = sqrt(g * h) is the speed at which a small-amplitude gravity wave travels across the water surface in shallow water of depth h. The Froude number is literally the ratio of the flow velocity to this wave speed. When V < c (Fr < 1) the flow moves slower than waves and is subcritical; when V > c (Fr > 1) the flow outruns its own waves and is supercritical.
Can the Froude number be used in pipe flow?
The Froude number as defined here applies to free-surface (open channel) flow, where a gravity restoring force acts on the water surface. For full-pipe pressure flow, gravity is not the dominant restoring mechanism and the Reynolds number is the primary dimensionless parameter. However, partially filled pipes and sewers do have a free surface and are analysed like open channels, with the Froude number used to check whether flow is subcritical or supercritical.