Froude Number Interactive Calculator

If you're working with open channel hydraulics—anything from spillways, stilling basins, ship hulls, or city storm drains—figure out your flow regime (subcritical, critical, or supercritical) before you start designing the channel geometry. This Froude Number Calculator gives you a quick way to find the Froude number, flow velocity, hydraulic depth, critical depth, critical velocity, or specific energy using velocity and depth. It’s relevant whether you’re looking at river crossings, hull performance, or flood risk. Get it wrong, and you risk undersized energy dissipators, unstable jumps, or erosion that’s expensive to correct. Below you’ll find the essential equations, a spillway example, practical regime notes, and answers to specific questions on wave work, jumps, and scaling models.

What is the Froude Number?

Froude number is a ratio that shows whether inertia or gravity drives the flow. If it’s less than 1, you’re looking at subcritical (slower, deeper) flow. Over 1, it’s supercritical (fast, shallow). Right at 1—that’s the dividing line, called critical flow.

Simple Explanation

The Froude number basically measures how your flow speed stacks up against the speed of surface waves in that fluid. If the water speed is lower than the wave speed, you get calm, deeper flow—that’s subcritical. Faster than the wave? Things turn rapid and turbulent (supercritical). You’ll use this in anything from sizing a river channel to predicting how a boat’s wake behaves.

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Open Channel Flow Diagram

How to Use This Calculator

1. Choose what you want to solve for—either Froude number, velocity, depth, or related parameter.
2. Enter the relevant numbers: velocity (m/s), hydraulic depth (m), Froude number, discharge (m³/s), or channel width (m) as needed for your mode.
3. Gravitational acceleration defaults to 9.81 m/s²; that should be fine for anything on Earth, but adjust if you’re working elsewhere.
4. Click Calculate and check your result.

Froude Number Calculator

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Governing Equations

The formula below is standard for Froude number calculations.

Here's how to calculate critical depth for a rectangular channel.

Here's how to calculate specific energy.

Simple numbers: If your velocity is 3.5 m/s, depth is 2.0 m, and gravity is 9.81 m/s², then Fr = 3.5 / √(9.81 × 2.0) = 3.5 / 4.429 = 0.790. That’s subcritical flow. Specific energy = 2.0 + (3.5²)/(2 × 9.81) = 2.625 m.

Theory & Practical Applications

In open channel flows, Froude number is the main check for regime, not Reynolds. Froude tells you if the flow’s inertia or gravity is calling the shots, and that determines if you get upstream influence, hydraulic jumps, or which direction disturbances move. This basic distinction sets everything from water surface profile to where you'll get energy dissipation.

Physical Significance and Wave Mechanics

The bottom of the Froude formula, √(g·h), gives you the surface wave speed. If your flow matches that speed (Fr = 1), you’re at critical—disturbances just hang in place. Less than that, and you get backwater: waves and changes in depth can push upstream. That’s why structures like gates or weirs can control water levels quite a way upstream from where they sit, especially in rivers or irrigation canals.

If you’re over that wave speed (Fr > 1), the flow is too fast—the water outruns its disturbances. Downstream conditions can’t reach back upstream. This is the territory of hydraulic jumps, standing waves, and sudden depth changes. There’s a direct analogy to Mach number in compressible flow: Fr = 1 in hydraulics is the equivalent of sonic speed in gas dynamics.

Critical Flow and Hydraulic Control Structures

Critical flow happens at control points—think weirs and flumes. At critical depth, for a given structure geometry, you get a set relationship between depth and flow that ignores what’s happening downstream. That’s why measuring flumes work reliably without being too sensitive to tailwater effects. But critical flow isn’t stable on its own. If energy or depth drifts slightly, the system jumps to subcritical or supercritical. For this reason, long sections of canal are normally kept away from critical by design; only short, controlled transitions are used.

Hydraulic Jump Formation and Energy Dissipation

If supercritical flow loses velocity (like hitting a flatter slope or a deeper pool), you’ll often get a hydraulic jump: abrupt, turbulent rise in water surface where a lot of energy is suddenly lost. The ratio of downstream to upstream depth in the jump comes straight from the upstream Froude number. Higher Froude, bigger jumps. The jump is useful—designers use it to bleed off destructive energy right below spillways or chutes. Get the basin size or sequent depth wrong, though, and the jump will either run downstream (too short a basin), or form irregularly (wrong sequent depth), causing problems instead of solving them.

Naval Architecture and Ship Hull Design

In ships, the Froude number—based on length—tells you when you hit “hull speed,” where wave drag starts to pile up. Most displacement hulls won’t do much better than Fr ≈ 0.5, no matter how much power you throw at them. Planing hulls get past this by skipping over the waves. If you’re testing ship models, you’ll need to match Froude number between model and full-size, which means scaling speed with the square root of the length ratio. You can’t match both Froude and Reynolds at the same time (unless you want an impossibly large tank), so you rely on Froude similarity and adjust for viscous drag as best you can.

River Engineering and Sediment Transport

Most rivers under normal flows have a Froude number of about 0.1 to 0.3. During floods, it can rise—but usually stays subcritical. If a river gets steep or narrows enough, sections go supercritical—that’s where you find hydraulic controls or steep drops. Sediment transport is highly sensitive to Froude. Since bedload increases rapidly with velocity, even a small jump in Fr causes a big jump in erosion or deposition—sometimes by orders of magnitude during floods.

Urban Drainage and Storm Sewer Design

Stormwater pipes and channels should be kept in subcritical flow to avoid jumps, surging, or erratic behavior in manholes. Typical design limits Fr to 0.85 or less. In steep terrain, sometimes supercritical flow is unavoidable—if so, you’ll need energy dissipation at the outlets. For culverts, knowing if the flow is inlet-controlled (supercritical in the barrel) or outlet-controlled (subcritical, depth set by downstream) is a big deal—if you get this call wrong in design, upstream flooding or road overtopping is a real risk.

Worked Example: Spillway Stilling Basin Design

Concrete spillway, with these specs:

• Unit discharge: q = 8.5 m³/s per meter
• Depth at spillway toe: h₁ = 0.75 m
• Channel width: b = 12 m (total Q = 102 m³/s)
• g = 9.81 m/s²

Step 1: Approach velocity
V₁ = q / h₁ = 8.5 / 0.75 = 11.33 m/s

Step 2: Approach Froude number
Fr₁ = 11.33 / √(9.81 × 0.75) = 11.33 / 2.715 = 4.173

This confirms strong supercritical flow—definitely needs a stilling basin downstream, or energy will cause damage.

Step 3: Sequent depth by momentum
h₂/h₁ = 0.5 × (√(1 + 8Fr₁²) - 1) = 0.5 × (√(1 + 8 × 4.173²) - 1) = 0.5 × (√140.2 - 1) = 0.5 × 10.84 = 5.42
h₂ = 5.42 × 0.75 = 4.065 m

Step 4: Downstream velocity
V₂ = q / h₂ = 8.5 / 4.065 = 2.091 m/s

Step 5: Downstream Froude
Fr₂ = 2.091 / √(9.81 × 4.065) = 2.091 / 6.314 = 0.331

Stable subcritical flow after jump. Energy dissipated as intended.

Step 6: Energy calculation
Before: E₁ = 0.75 + 11.33²/(2×9.81) = 0.75 + 6.539 = 7.289 m
After: E₂ = 4.065 + 2.091²/(2×9.81) = 4.065 + 0.223 = 4.288 m
ΔE = 3.001 m lost (about 41.2% energy loss at the jump—typical in these situations).

Step 7: Stilling basin length
For this Fr₁ (about 4.2), practical designs use length ≈ 5 × h₂ = 5 × 4.065 = 20.3 m. Designers usually call for a bit more—say, 22 m—and include blocks to anchor the jump and improve energy dissipation.

So getting the Froude number right upfront is critical: it sets the foundation for basin size, jump properties, and whether energy will go where you want or scour something you didn’t mean to touch.

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Frequently Asked Questions

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