The Froude Number — Inertia vs. Gravity in Free-Surface Flows
“The Froude number is the Mach number of water. It tells you whether information can travel upstream — and that single fact determines the entire character of a free-surface flow.”
1. Introduction
Whenever a fluid flows with a free surface — a boundary between liquid and gas — gravity enters as a restoring force that drives surface waves. The competition between the inertia of the flowing fluid and the gravitational restoring force is quantified by the Froude number (Fr), named in honour of the Victorian naval engineer William Froude.
The Froude number answers a deceptively simple question: Is the flow faster than the waves it generates? The answer divides all free-surface flows into two fundamentally different regimes — subcritical and supercritical — separated by a critical condition at Fr = 1 that produces one of the most spectacular phenomena in fluid mechanics: the hydraulic jump.
Definition and Parameters
$$Fr = rac{v}{sqrt{gL}}$$
- $v$ — characteristic flow velocity (m/s)
- $g$ — gravitational acceleration, 9.81 m/s²
- $L$ — characteristic length: flow depth $h$ for open channels; hull length $L$ for ships
All three quantities carry SI units that cancel exactly: $[v/sqrt{gL}] = ext{m s}^{-1}/ ext{m s}^{-1} = 1$. The Froude number is dimensionless.
2. Historical Background
William Froude (1810–1879) was a British civil and naval engineer who confronted a practical problem: the Royal Navy needed to predict the wave-making resistance of full-scale warships from small model experiments. Froude reasoned that a model ship and its prototype produce geometrically similar wave patterns only when $v/sqrt{gL}$ is identical in both. He built the world’s first ship model testing facility at Torquay in 1871. His son, Robert Edmund Froude, formalised the eponymous number.
3. Physical Interpretation — The Gravity-Wave Analogy
Long-wavelength gravity waves in shallow water of depth $h$ propagate at the celerity:
$$c = sqrt{gh}$$
The Froude number is therefore the ratio of flow velocity to gravity-wave celerity: $Fr = v/c$. This is exactly analogous to the Mach number $ ext{Ma} = v/a$ in gas dynamics. The governing equations for shallow-water flow are mathematically identical to those for one-dimensional compressible gas flow:
Froude–Mach Analogy
| Gas Dynamics | Shallow-Water Flow |
|---|---|
| Speed of sound $a = sqrt{gamma RT}$ | Wave celerity $c = sqrt{gh}$ |
| Mach number Ma = v/a | Froude number Fr = v/c |
| Subsonic (Ma < 1) | Subcritical (Fr < 1) |
| Sonic / choked (Ma = 1) | Critical flow (Fr = 1) |
| Supersonic (Ma > 1) | Supercritical (Fr > 1) |
| Normal shock wave | Hydraulic jump |
4. The Three Flow Regimes
| Regime | Fr | Characteristics | Examples |
|---|---|---|---|
| Subcritical (tranquil) | Fr < 1 | Deep, slow; waves propagate upstream; controlled from downstream | Reservoir, gentle rivers |
| Critical | Fr = 1 | Transition; minimum specific energy for given discharge | Weir crest, jump location |
| Supercritical (rapid) | Fr > 1 | Shallow, fast; waves swept downstream; controlled from upstream | Spillway chute, rapids |
5. The Hydraulic Jump
A hydraulic jump occurs when a supercritical flow (Fr₁ > 1) abruptly transitions to subcritical flow (Fr₂ < 1). It is the shallow-water analogue of a normal shock wave: a thin, highly dissipative transition.
Applying the momentum equation across the jump yields the Bélanger equation (1828):
$$ rac{h_2}{h_1} = rac{1}{2}left(sqrt{1 + 8Fr_1^2} – 1
ight)$$
Worked Example 1: Hydraulic Jump Below a Sluice Gate
Water issues from a sluice gate: $h_1 = 0.25$ m, $v_1 = 7.5$ m/s.
(a) Inlet Froude number: $Fr_1 = 7.5/sqrt{9.81 imes 0.25} = 7.5/1.565 = 4.79$
(b) Conjugate depth:
$$h_2 = rac{0.25}{2}(sqrt{1 + 8 imes4.79^2}-1) = 0.125(sqrt{184.7}-1) = 0.125 imes 12.59 = 1.57 ext{ m}$$
(c) Energy dissipated: $E_1 = 0.25 + 7.5^2/(2 imes9.81) = 3.12$ m; $v_2 = v_1h_1/h_2 = 1.19$ m/s; $E_2 = 1.57 + 1.19^2/19.62 = 1.64$ m; $Delta E = 1.48$ m (47% loss). Engineers use this in stilling basins to prevent scour.
6. Ship Hull Speed
A displacement hull moving at speed $v$ generates a bow wave of wavelength $lambda = 2pi v^2/g$. When $lambda = L$ (hull length), the ship rides in its own trough and wave-making resistance peaks sharply. This “hull speed” corresponds to:
$$Fr_{hull} = rac{v_{hull}}{sqrt{gL}} = rac{1}{sqrt{2pi}} pprox 0.398$$
Worked Example 2: Container Ship at Fr = 0.22
Hull length $L = 300$ m. Service speed at $Fr = 0.22$:
$$v = 0.22sqrt{9.81 imes 300} = 0.22 imes 54.24 = 11.93 ext{ m/s} pprox 23.2 ext{ knot}$$
Hull speed: $v_{hull} = 0.398 imes 54.24 = 21.6$ m/s = 42 knot. Large containerships operate well below hull speed (Fr ≈ 0.18–0.25) to minimise wave drag.
7. Froude Similarity for Scale Models
For free-surface flows, model and prototype must share the same Froude number:
$$ rac{v_m}{v_p} = sqrt{lambda_r}, quad rac{Q_m}{Q_p} = lambda_r^{5/2}, quad rac{t_m}{t_p} = sqrt{lambda_r}$$
where $lambda_r = L_m/L_p$ is the geometric scale ratio.
Worked Example 3: Spillway Model at 1:50 Scale
Prototype flood discharge $Q_p = 8{,}000$ m³/s, $lambda_r = 1/50$.
$$Q_m = 8000 imes (1/50)^{2.5} = 8000/17{,}678 = 0.453 ext{ m}^3 ext{/s}$$
Prototype velocity 12 m/s scales to: $v_m = 12/sqrt{50} = 1.70$ m/s — easily achievable in a laboratory.
8. Froude Number in Biomechanics
Alexander (1984) showed that all terrestrial mammals transition from walking to running at a universal Froude number $Fr = v^2/(gL) pprox 0.25$ (here $L$ is leg length). This law holds from mice to elephants and was used to estimate dinosaur speeds from fossil trackways.
Worked Example 4: Human Gait Transition
Leg length $L = 0.9$ m. Predicted walk-to-run transition speed:
$$v = sqrt{0.25 imes 9.81 imes 0.9} = sqrt{2.207} = 1.49 ext{ m/s} pprox 5.4 ext{ km/h}$$
Observed human gait transition: 5–6 km/h. Agreement is excellent.
9. Breaking Waves
As ocean waves shoal onto a beach, wave celerity decreases as $c = sqrt{gh}$ while period is conserved. Height increases to conserve energy flux. Breaking occurs when the local Froude number at the crest exceeds 1 (equivalently, when $H/h pprox 0.78$, the McCowan breaking criterion). This governs surf-zone dynamics and coastal structure design.
10. Densimetric Froude Number
In stratified flows (estuaries, fjords, atmospheric gravity currents), density variation replaces full gravity with reduced gravity $g’ = g(
ho_1-
ho_2)/
ho_2$:
$$Fr_d = rac{v}{sqrt{g’ h}}$$
When $Fr_d > 1$ the dense layer is swept downstream; when $Fr_d < 1$ it can propagate upstream. This controls salt-wedge intrusion, lava flows, and pyroclastic density currents.
11. The Kelvin Wake Angle
Lord Kelvin (1887) proved that a point source moving at Fr < 0.4 on a free surface generates a V-shaped wake with half-angle $ heta = rcsin(1/3) pprox 19.47°$, independent of speed. At Fr > 0.4 the angle narrows — a qualitative change visible in satellite imagery of fast ships.
12. IYPT 2027 Connections
IYPT 2027 Key Links
Problem P7 — Sinking Funnel: Fr at the funnel throat determines whether the air-water interface forms a smooth taper or collapses into droplets. Transition near Fr ≈ 1.
Wave propagation problems: Phase speed, group velocity, and wave-obstacle interaction all require Fr to determine whether reflected waves can travel upstream and affect the source region.
Rapidly varied flows: Hydraulic jumps, apron flows, and weir overflows all governed by Fr and the Bélanger equation.
13. Key Equations Summary
Froude Number Toolkit
- Definition: $Fr = v/sqrt{gL}$
- Gravity wave speed: $c = sqrt{gh}$
- Bélanger equation: $h_2/h_1 = rac{1}{2}(sqrt{1+8Fr_1^2}-1)$
- Hull speed: $Fr_{hull} pprox 0.398$
- Froude scaling — velocity: $v_m/v_p = sqrt{lambda_r}$
- Froude scaling — discharge: $Q_m/Q_p = lambda_r^{5/2}$
- Gait transition: $Fr = v^2/(gL) pprox 0.25$
- Kelvin angle: $ heta pprox 19.47°$ for Fr < 0.4
14. Flow Regime Quick-Reference
| Application | Typical Fr | Notes |
|---|---|---|
| Supertanker | 0.12–0.18 | Minimum wave resistance |
| Naval destroyer | 0.30–0.45 | Near hull speed |
| Racing sailboat | 0.5–1.5 | Partial to full planing |
| Jet-ski | 2–5 | Full planing |
| Mountain rapid | 1.0–3.0 | Supercritical |
| Hydraulic jump inlet | 1.5–6 | Up to 75% energy dissipated |
| Human walking | 0.15–0.25 | Inverted-pendulum gait |
| T. rex (fossil tracks) | ~0.25 | Universal gait-transition law |
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