Froude Number Interactive Calculator

The Froude number is a dimensionless parameter that characterizes the ratio of inertial forces to gravitational forces in open channel flow and wave-making resistance. Critical in naval architecture, hydraulic engineering, and coastal design, it determines whether flow is subcritical, critical, or supercritical — governing wave behavior, energy dissipation, and sediment transport. Engineers use Froude number analysis to predict wave formation around ship hulls, design spillways and weirs, and assess flooding risk in urban drainage systems.

📐 Browse all free engineering calculators

Open Channel Flow Diagram

Froude Number Calculator

Governing Equations

Theory & Practical Applications

The Froude number represents one of the most consequential dimensionless parameters in fluid mechanics, governing the transition between fundamentally different flow regimes in free-surface flows. Unlike pipe flow where Reynolds number dominates, open channel flow behavior is primarily controlled by the balance between inertial forces (represented by flow velocity) and gravitational forces (represented by wave propagation speed). This balance determines whether disturbances can propagate upstream, whether hydraulic jumps form, and how energy dissipates along the channel.

Physical Significance and Wave Mechanics

The denominator √(g·h) represents the celerity of shallow-water gravity waves — the speed at which small disturbances propagate on the water surface. When flow velocity equals this wave speed (Fr = 1), the system reaches critical flow where disturbances remain stationary relative to the flow. In subcritical flow (Fr less than 1), waves can propagate upstream, allowing backwater effects from downstream obstructions to influence upstream conditions. This is why gates, weirs, and changes in channel geometry affect water surface profiles extending far upstream in rivers and canals.

In supercritical flow (Fr greater than 1), the flow moves faster than disturbance propagation speed, creating a regime where upstream conditions cannot be influenced by downstream geometry. This leads to the formation of standing waves, hydraulic jumps when transitioning back to subcritical flow, and severe erosion potential. The analogy to supersonic aerodynamics is direct — the Froude number in hydraulics plays the same role as the Mach number in compressible gas dynamics, with Fr = 1 corresponding to sonic flow.

Critical Flow and Hydraulic Control Structures

Critical flow (Fr = 1) occurs naturally at hydraulic control points such as weirs, flumes, and channel contractions. Engineers exploit this phenomenon in flow measurement — at critical depth, there exists a unique relationship between discharge and depth that is independent of downstream conditions. Parshall flumes and broad-crested weirs operate on this principle, providing accurate flow measurements without the head loss and sediment accumulation problems of traditional sharp-crested weirs.

However, critical flow represents an unstable equilibrium. The specific energy curve shows that for a given energy level, two possible depths exist — subcritical (deep and slow) and supercritical (shallow and fast). At minimum energy, these two depths converge to critical depth. Any perturbation at critical flow tends to push the system toward either subcritical or supercritical conditions, which is why critical flow is intentionally avoided in canal design except at measurement and control structures.

Hydraulic Jump Formation and Energy Dissipation

When supercritical flow encounters an obstruction or enters a milder slope forcing transition to subcritical conditions, a hydraulic jump forms — an abrupt rise in water surface accompanied by intense turbulence and energy dissipation. The sequent depth ratio (depth after jump to depth before jump) depends directly on the upstream Froude number. For Fr = 2, the sequent depth ratio is approximately 2.25; for Fr = 5, it reaches 6.86. This relationship allows engineers to design stilling basins below spillways where controlled hydraulic jumps dissipate energy that would otherwise cause severe downstream erosion.

The energy loss in a hydraulic jump increases dramatically with Froude number. At Fr = 2, approximately 17% of upstream specific energy is dissipated; at Fr = 5, this increases to 56%. This makes hydraulic jumps highly effective energy dissipators, but also means that improper stilling basin design (insufficient length, incorrect sequent depth) can result in the jump being swept out of the basin, transferring the erosion problem downstream rather than solving it.

Naval Architecture and Ship Hull Design

In naval architecture, the Froude number determines wave-making resistance — the energy lost to generating surface waves as a vessel moves through water. For displacement hulls, wave resistance becomes dominant at Froude numbers (based on hull length rather than depth) above 0.4, creating a "hull speed" barrier where additional power produces minimal speed increase. This is why displacement vessels rarely exceed Fr ≈ 0.5, while planing hulls transition to skimming over the surface at higher Froude numbers.

Scale model testing in towing tanks relies on Froude number similarity — to accurately predict full-scale wave resistance, the model must operate at the same Froude number as the prototype. Since gravitational acceleration cannot be scaled, this requires velocity to scale with the square root of the length ratio. A 1:50 scale model must be towed at 1/√50 ≈ 1/7.07 of prototype speed to maintain Froude similarity. Reynolds number cannot be simultaneously matched at these scales, requiring empirical corrections for viscous effects.

River Engineering and Sediment Transport

In natural rivers, the Froude number typically ranges from 0.1 to 0.3 during normal flow, rising to 0.5-0.8 during floods. This subcritical regime allows gradual adjustment of water surface profiles to downstream conditions. However, reaches with steep gradients or constrictions can develop supercritical flow, creating hydraulic control points that fix the stage-discharge relationship regardless of downstream conditions.

Sediment transport intensifies dramatically as Froude number increases. Bed shear stress scales with velocity squared, but sediment transport capacity often scales with velocity to the third, fourth, or even fifth power depending on grain size and flow conditions. A river reach transitioning from Fr = 0.2 to Fr = 0.8 during a flood event can experience sediment transport rates increasing by a factor of 100 or more, explaining the extreme erosion and deposition observed during major floods.

Urban Drainage and Storm Sewer Design

Storm sewer systems must be designed to operate in subcritical flow under all design conditions to prevent hydraulic jumps, surcharging, and geyser formation at manholes. Design guidelines typically require Fr ≤ 0.85 to maintain adequate freeboard and flow stability. When supercritical flow does occur — often due to steep gradients in mountainous terrain — energy dissipation structures must be provided at manholes to prevent downstream pipe damage.

The relationship between Froude number and flow regime becomes critical in culvert design. A culvert operating under inlet control (supercritical flow in the barrel) behaves completely differently from one under outlet control (subcritical flow backed up by downstream conditions). Misidentifying the control regime during design can result in severe underestimation of headwater elevation, leading to roadway overtopping or upstream flooding.

Worked Example: Spillway Stilling Basin Design

Consider a concrete spillway with the following conditions:

• Discharge per unit width: q = 8.5 m³/s per meter
• Flow depth at toe of spillway: h₁ = 0.75 m
• Channel width: b = 12 m (total discharge Q = 8.5 × 12 = 102 m³/s)
• Gravitational acceleration: g = 9.81 m/s²

Step 1: Calculate approach velocity

V₁ = q / h₁ = 8.5 m³/s/m / 0.75 m = 11.33 m/s

Step 2: Calculate approach Froude number

Fr₁ = V₁ / √(g·h₁) = 11.33 / √(9.81 × 0.75) = 11.33 / 2.715 = 4.173

This high Froude number confirms supercritical flow requiring energy dissipation. The flow is classified as "choppy" to "steady jump" range based on standard hydraulic jump classification.

Step 3: Calculate sequent depth using momentum equation

The sequent depth ratio for a hydraulic jump is given by:

h₂/h₁ = 0.5 × (√(1 + 8Fr₁²) - 1)

h₂/h₁ = 0.5 × (√(1 + 8 × 4.173²) - 1) = 0.5 × (√140.2 - 1) = 0.5 × 10.84 = 5.42

h₂ = 5.42 × 0.75 m = 4.065 m

Step 4: Calculate velocity after jump

V₂ = q / h₂ = 8.5 / 4.065 = 2.091 m/s

Step 5: Verify subcritical flow after jump

Fr₂ = V₂ / √(g·h₂) = 2.091 / √(9.81 × 4.065) = 2.091 / 6.314 = 0.331

Confirmed subcritical (Fr₂ less than 1), indicating successful energy dissipation.

Step 6: Calculate energy dissipation

Specific energy before jump: E₁ = h₁ + V₁²/(2g) = 0.75 + 11.33²/(2×9.81) = 0.75 + 6.539 = 7.289 m

Specific energy after jump: E₂ = h₂ + V₂²/(2g) = 4.065 + 2.091²/(2×9.81) = 4.065 + 0.223 = 4.288 m

Energy loss: ΔE = E₁ - E₂ = 7.289 - 4.288 = 3.001 m

Percentage dissipated: (ΔE/E₁) × 100% = (3.001/7.289) × 100% = 41.2%

Step 7: Determine stilling basin length

For Fr₁ = 4.17, empirical relationships suggest basin length L ≈ 5 × h₂ = 5 × 4.065 = 20.3 m. Conservative design would specify L = 22 m with baffle blocks to stabilize the jump position and chute blocks at the entrance to promote jump formation.

This analysis demonstrates how Froude number governs every aspect of hydraulic jump design — from predicting whether a jump will form, to calculating sequent depth, to sizing the stilling basin. An error in the initial Froude number calculation propagates through all subsequent design steps, potentially resulting in a basin too short to contain the jump or designed for the wrong flow regime entirely.

For additional fluid dynamics calculations and analysis tools, visit the complete engineering calculator library.

Frequently Asked Questions