What length should I use in Fr = V/sqrt(gL)?

Use hydraulic depth A/T for open channels (or y in rectangular sections) and waterline length for ships. Choose L that represents the relevant wave celerity.

What does Fr = 1 mean?

The flow speed equals shallow-water wave speed; the flow is critical and highly sensitive to control. Many measuring flumes operate near this condition.

Why match Froude number in models?

Matching Fr preserves free-surface dynamics and wave patterns between model and prototype, enabling valid scale predictions of water surface profiles and hydraulic jumps.

Can a flow switch between subcritical and supercritical?

Yes. Constrictions or steep slopes can push Fr above 1 (supercritical). Returning to subcritical typically occurs through a hydraulic jump that dissipates energy.

Froude Number Calculator: Calculate Fr for Fluid Flow Analysis

Q: How is Froude different from Reynolds?

Froude compares inertia to gravity/wave effects; Reynolds compares inertia to viscosity. Use Fr for free-surface behavior and wave propagation, Re for turbulence and head loss.

Froude’s Number Calculator

Froude’s Number: The Complete Guide

Froude’s number is a cornerstone of free-surface hydraulics and naval architecture. It compares a flow’s inertial effects to gravity effects and signals whether surface waves can propagate upstream. In symbols, the Froude number is the ratio of flow speed to the characteristic wave speed induced by gravity. Your calculator above can compute Fr, classify subcritical/supercritical states, and solve related design questions such as critical depth, hydraulic jumps, and scale modeling. This page explains the equations behind those results, variables and units, when to use each form, worked examples, practical interpretations, and common pitfalls.

\( \displaystyle \textbf{Generic definition:}\quad \mathrm{Fr} = \frac{V}{\sqrt{g\,L}} \)

\( \displaystyle \textbf{Open-channel (shallow-water) form:}\quad \mathrm{Fr} = \frac{V}{\sqrt{g\,d_h}} ,\qquad d_h=\frac{A}{T} \ \text{(hydraulic depth; }A=\text{area},\,T=\text{top width)} \)

\( \displaystyle \textbf{Ship hydrodynamics (deep water) form:}\quad \mathrm{Fr} = \frac{U}{\sqrt{g\,L_{\mathrm{WL}}}} \)

The three canonical regimes are: subcritical (\(\mathrm{Fr}<1\), gravity/waves dominate and can travel upstream), critical (\(\mathrm{Fr}=1\)), and supercritical (\(\mathrm{Fr}>1\), inertia dominates; information and small perturbations move downstream only).

Core Equations & When to Use Each

Open-channel flows (rivers, canals, spillways): use \( \mathrm{Fr} = V/\sqrt{g\,d_h} \). Here \(d_h=A/T\) captures depth relevant to long gravity waves.
Rectangular channels: \( d_h = y \) (flow depth), so \( \mathrm{Fr}=V/\sqrt{g\,y} \).
Ship/boat resistance & waves: use \( \mathrm{Fr} = U/\sqrt{g\,L_{\mathrm{WL}}} \) with waterline length \(L_{\mathrm{WL}}\).
Rapidly varied flow (hydraulic jumps): Fr determines sequent depth and energy dissipation; upstream state \( \mathrm{Fr}_1 \) drives jump strength.
Dynamic similitude in modeling: scale models should match Froude number to reproduce wave/gravity behavior.

\( \displaystyle \textbf{Critical state:}\quad \mathrm{Fr}=1 \quad\Rightarrow\quad V = c = \sqrt{g\,d_h} \)

At critical conditions, the mean flow speed equals the shallow-water wave celerity \(c\). Around control structures, designers often seek critical conditions intentionally (e.g., flumes) to stabilize measurements or transitions.

Variables, Units & Notation

\(V\) (or \(U\)): mean flow speed (m/s).
\(g\): gravitational acceleration, \(9.80665\,\text{m/s}^2\).
\(L\): characteristic length for wave speed (m)—context dependent.
\(d_h\): hydraulic depth \(=A/T\) (m). For a rectangular channel \(d_h=y\).
\(A\): flow area (m\(^2\)). \(T\): top width (m). \(y\): depth (m).
\(L_{\mathrm{WL}}\): waterline length of a hull (m).
\(q\): discharge per unit width (m\(^2\)/s) in wide or rectangular channels.

Application	Length \(L\)	Why this choice?
Open-channel flow	\(d_h=A/T\)	Controls long gravity-wave speed \(c=\sqrt{g\,d_h}\).
Rectangular channel	\(y\)	Hydraulic depth equals geometric depth.
Ship hydrodynamics	\(L_{\mathrm{WL}}\)	Dominant wave-making scales with waterline length.
Spillways/weirs	local depth \(y\)	Wave celerity tied to local flow thickness.

How to Calculate Froude’s Number (Step-by-Step)

Pick the right \(L\): For open-channel work use \(d_h=A/T\); for simple rectangular sections use \(y\); for ships use \(L_{\mathrm{WL}}\).
Compute wave speed: \( c=\sqrt{g\,L} \) (or \( \sqrt{g\,d_h} \)).
Measure/compute mean velocity: \( V = Q/A \) for channels, or craft speed \(U\) for vessels.
Form the ratio: \( \mathrm{Fr}=V/c \).
Classify regime: \( \mathrm{Fr}<1 \) subcritical; \(=1\) critical; \(>1\) supercritical.
Apply to decisions: check backwater influence, hydraulic jump need/strength, structure placement, or wave-making resistance.

Your calculator can automate these steps, switching among open-channel, rectangular, and ship-hull conventions and reporting the regime with clear guidance.

Worked Examples

Example 1 — Rectangular Channel Classification

Given: Rectangular channel, depth \(y=1.20\,\text{m}\), discharge \(Q=6.0\,\text{m}^3/\text{s}\), width \(b=3.0\,\text{m}\).

\( \displaystyle A=b\,y=3.0\times1.20=3.60\,\text{m}^2,\quad V=Q/A=6.0/3.60=1.667\,\text{m/s} \)

\( \displaystyle \mathrm{Fr}=\frac{V}{\sqrt{g\,y}} =\frac{1.667}{\sqrt{9.80665\times1.20}} \approx \frac{1.667}{3.432} \approx 0.486 \)

Answer: \(\mathrm{Fr}\approx 0.49 < 1\) → subcritical; waves can travel upstream.

Example 2 — Finding Critical Depth (Rectangular Channel)

Given: Unit discharge \( q=Q/b = 2.0\,\text{m}^2/\text{s} \). For a rectangular channel, the critical condition satisfies \( \mathrm{Fr}=1 \Rightarrow V^2=g\,y_c \) and \( V=q/y \).

\( \displaystyle \mathrm{Fr}=1:\quad \frac{(q/y_c)^2}{g\,y_c}=1 \Rightarrow \frac{q^2}{g\,y_c^3}=1 \Rightarrow y_c=\left(\frac{q^2}{g}\right)^{1/3} \)

\( \displaystyle y_c=\left(\frac{2.0^2}{9.80665}\right)^{1/3} =\left(\frac{4}{9.80665}\right)^{1/3} \approx (0.408)^{1/3}\approx 0.748\,\text{m} \)

Answer: \( y_c \approx 0.75\,\text{m} \). If actual depth \(y>y_c\), flow is subcritical; if \(y<y_c\), supercritical.

Example 3 — Hydraulic Jump Strength from Upstream \(\mathrm{Fr}_1\)

Given: A supercritical stream with depth \(y_1=0.30\,\text{m}\) and velocity \(V_1=6.0\,\text{m/s}\) (rectangular channel). Compute \(\mathrm{Fr}_1\) and the sequent depth \(y_2\).

\( \displaystyle \mathrm{Fr}_1 = \frac{V_1}{\sqrt{g\,y_1}} = \frac{6.0}{\sqrt{9.80665\times 0.30}} = \frac{6.0}{\sqrt{2.942}} \approx \frac{6.0}{1.715} \approx 3.50 \)

For a rectangular hydraulic jump, the sequent (downstream) depth is

\( \displaystyle y_2 = \frac{y_1}{2}\left(\sqrt{1+8\,\mathrm{Fr}_1^2}-1\right) = \frac{0.30}{2}\left(\sqrt{1+8\times 3.50^2}-1\right) \approx 0.15(\sqrt{1+98}-1) \approx 0.15(9.95-1) \approx 1.34\,\text{m} \)

Answer: \(\mathrm{Fr}_1\approx 3.5\) (strongly supercritical); jump raises depth to \(y_2\approx 1.34\,\text{m}\) with substantial energy dissipation.

Example 4 — Boat Speed and Froude Number

Given: Small craft with waterline length \( L_{\mathrm{WL}}=6.5\,\text{m} \) travels at \( U=7.5\,\text{m/s} \) (≈ 14.6 knots).

\( \displaystyle \mathrm{Fr} = \frac{U}{\sqrt{g\,L_{\mathrm{WL}}}} = \frac{7.5}{\sqrt{9.80665 \times 6.5}} \approx \frac{7.5}{7.988} \approx 0.94 \)

Interpretation: \(\mathrm{Fr}\approx 0.94\) is near the “hull-speed” region for displacement vessels; wave-making resistance surges.

How to Interpret Froude’s Number

Subcritical (\(\mathrm{Fr}<1\)): Gravity/waves dominate; disturbances can move upstream. Backwater effects from downstream controls (dams, gates) can propagate upstream.
Critical (\(\mathrm{Fr}=1\)): Transition point; flow is highly sensitive; commonly used in flow-measurement structures (critical-depth flumes).
Supercritical (\(\mathrm{Fr}>1\)): Inertia dominates; information travels downstream. Flow is shallow and fast; hydraulic jumps may be required to dissipate energy safely.
Wave celerity link: \( c=\sqrt{g\,d_h} \) (shallow water). If \(V>c\), waves cannot travel upstream.
Model similitude: Matching \(\mathrm{Fr}\) between model and prototype preserves free-surface dynamics and wave patterns.

Common Use Cases

Channel design & flood control: classify reaches; anticipate backwater curves and jump locations.
Energy dissipation: design stilling basins by targeting hydraulic jump strengths via upstream \(\mathrm{Fr}_1\).
Spillways & weirs: place transitions to avoid unwanted supercritical lengths or air entrainment.
Harbors & boating: estimate wave-making and operational limits as \(\mathrm{Fr}\) approaches unity.
Scale models: ensure Froude similitude to reproduce prototype free-surface behavior in laboratory flumes or towing tanks.

Assumptions, Limitations & Pitfalls

Choice of \(L\): Pick a physically meaningful length. Using depth for free-surface shallow flows and waterline length for ships is not interchangeable.
Not a viscosity measure: Froude ignores viscosity; pair it with Reynolds number (\(\mathrm{Re}\)) when boundary-layer or head-loss issues matter.
Surface tension at small scales: When capillarity matters (thin sheets, small lab scales), include Weber number (\(\mathrm{We}\)); Froude alone won’t capture capillary waves.
Rapid geometry changes: Locally varied depths make a single \(\mathrm{Fr}\) ambiguous; evaluate section-by-section.
Non-Newtonian fluids: For slurries or mudflows, shallow-water wave speeds may deviate from \( \sqrt{g\,d_h} \).
Deep-water ship waves: The \(L_{\mathrm{WL}}\)-based Fr is a proxy; hull shape and interference patterns still matter for resistance.

FAQ: Froude’s Number

How is Froude different from Reynolds?

\(\mathrm{Fr}\) compares inertia to gravity (wave effects), while \(\mathrm{Re}\) compares inertia to viscous forces. Use \(\mathrm{Fr}\) for free-surface/wave behavior; use \(\mathrm{Re}\) for turbulence, drag, and head loss.

What length should I use in \(\mathrm{Fr}=V/\sqrt{gL}\)?

Use hydraulic depth \(d_h\) for open channels, geometric depth \(y\) for rectangular channels, and \(L_{\mathrm{WL}}\) for ship hydrodynamics. The right choice reflects the wave celerity of interest.

What does \(\mathrm{Fr}=1\) physically mean?

The mean flow speed equals the gravity-wave speed. Flow transitions are sensitive; small controls can have large effects. Many measuring devices exploit this state.

Why do we match Froude number in hydraulic models?

To preserve dynamic similarity of free-surface behavior and wave patterns. When Froude is matched, water surface profiles and jumps scale correctly (viscosity may still scale imperfectly).

Can a flow change from subcritical to supercritical?

Yes, via constrictions or slopes. The reverse transition typically occurs through a hydraulic jump, dissipating energy and thickening the flow.

Quick Checklist for Accurate Froude Calculations

Compute \(A, T\) accurately; set \(d_h=A/T\) (or \(y\) for rectangular sections).
Use consistent SI units; \(g=9.80665\,\text{m/s}^2\).
Evaluate \(\mathrm{Fr}\) section-by-section in nonuniform channels.
For scale models, match \(\mathrm{Fr}\); note that matching \(\mathrm{Re}\) simultaneously is generally impossible—document the compromise.
Check for capillary effects in very small scales (consider \(\mathrm{We}\)).
Use \(\mathrm{Fr}_1\) to size stilling basins and estimate sequent depths in hydraulic jumps.

Bottom Line

Froude’s number distills the interplay of inertia and gravity in free-surface flows and wave-making. With \( \mathrm{Fr}=V/\sqrt{gL} \), you can classify regimes, predict upstream influence, design energy dissipation, and achieve reliable similitude in models. Use the calculator above to compute \(\mathrm{Fr}\) quickly for channels or hulls, and return to this guide to choose the correct length scale, interpret results, and avoid common traps. Mastering Froude unlocks confident decisions in rivers, spillways, stormwater systems, laboratories, and naval architecture.