Froude Number Calculator
Calculate Froude number, velocity, hydraulic depth, flow rate, or critical velocity for open-channel flow with live flow regime interpretation.
Calculator is for informational purposes only. Terms and Conditions
Choose what to solve for
Select the unknown and input method. Required fields update automatically.
Enter the known values
Use hydraulic depth \(D=A/T\) for non-rectangular open channels.
Visual Check
See the open-channel relationship and where the result falls on the flow regime scale.
Solution
Live result, quick checks, warnings, and full solution steps.
Quick checks
- Flow regime—
Show solution steps See the equation, substitutions, assumptions, and result path
- Enter values to see the full solution steps and checks.
Source, Standards, and Assumptions
Calculation basis, constants, assumptions, and limitations.
Uses the standard open-channel Froude number relationship \(Fr=V/\sqrt{gD}\), where hydraulic depth is \(D=A/T\).
- Assumptions will appear after a valid calculation.
On this page
Calculator Guide
How to Use the Froude Number Calculator
The Froude Number Calculator above calculates whether flow is subcritical, near critical, or supercritical using \(Fr=V/\sqrt{gD}\). For open-channel flow, the most important detail is using hydraulic depth \(D=A/T\), not hydraulic radius.
Use the tool for quick open-channel flow checks, homework problems, flume and channel calculations, hydraulic jump screening, and model-similarity estimates. The result is dimensionless, so unit consistency matters more than the final unit label.
Quick Answer
Froude number compares flow velocity to gravity wave speed. If \(Fr<1\), flow is subcritical and usually slower and deeper. If \(Fr\approx1\), flow is critical. If \(Fr>1\), flow is supercritical and usually faster and shallower. In open-channel problems, calculate hydraulic depth with \(D=A/T\), then use \(Fr=V/\sqrt{gD}\).
Do not rely on a simplified Froude number alone when…
Do not use a single Froude number result as the only basis for final hydraulic design, flood modeling, spillway design, culvert performance, erosion protection, stilling basin sizing, or field acceptance. Final design should also check channel geometry, roughness, energy losses, hydraulic jumps, sediment movement, tailwater conditions, and project-specific requirements.
Inputs and Outputs Used by the Calculator
The calculator typically uses velocity, gravity, and hydraulic depth or characteristic length. In open-channel geometry mode, it may calculate velocity from discharge and area, then calculate hydraulic depth from area and top width.
| Type | Value | What It Means | Common Unit |
|---|---|---|---|
| Input | Velocity, \(V\) | Average flow velocity through the section or characteristic velocity for the problem. | m/s, ft/s |
| Input | Gravity, \(g\) | Acceleration due to gravity. Use a consistent unit system with velocity and depth. | 9.80665 m/s² or 32.174 ft/s² |
| Input | Hydraulic Depth, \(D\) | Open-channel depth term equal to flow area divided by top width. | m, ft |
| Input | Discharge, \(Q\) | Flow rate used to calculate velocity when \(V=Q/A\). | m³/s, ft³/s |
| Input | Flow Area, \(A\) | Cross-sectional area of flowing water normal to the mean velocity. | m², ft² |
| Input | Top Width, \(T\) | Free-surface width used to calculate hydraulic depth \(D=A/T\). | m, ft |
| Output | Froude Number, \(Fr\) | Dimensionless ratio of inertial effects to gravity-wave effects. | unitless |
| Output | Flow Regime | Classification as subcritical, near critical, or supercritical. | text result |
| Output | Critical Velocity, \(V_c\) | Velocity where \(Fr=1\) for the entered hydraulic depth. | m/s, ft/s |
Froude Number Formula
The most common Froude number formula divides velocity by the shallow-water wave speed. For open-channel flow, use hydraulic depth \(D=A/T\) unless a specific problem defines a different characteristic length.
Main Froude Number Formula
Use this form when velocity \(V\), gravitational acceleration \(g\), and hydraulic depth or characteristic length \(D\) are known.
Hydraulic Depth for Open-Channel Flow
Hydraulic depth is flow area divided by top width. It is not the same as hydraulic radius \(R=A/P\).
Froude Number from Discharge, Area, and Top Width
This form is useful when discharge \(Q\), flow area \(A\), and top width \(T\) are known instead of velocity and hydraulic depth directly.
Useful Rearranged Forms
These rearranged forms help solve for velocity, hydraulic depth, or critical velocity when the Froude number is known.
Why the formula matters
Froude number controls how disturbances, surface waves, and downstream conditions interact with the flow. A result near 1 is especially important because small changes in depth, slope, or discharge can shift the channel between subcritical and supercritical behavior.
What the Variables Mean
Each variable must represent the same flow condition and unit system. The most common mistake is entering hydraulic radius where the formula requires hydraulic depth.
| Symbol | Meaning | How to Enter It |
|---|---|---|
| \(Fr\) | Froude number, a dimensionless flow-regime parameter. | Enter directly only when solving for velocity, depth, or gravity. |
| \(V\) | Mean flow velocity. | Use \(V=Q/A\) when discharge and cross-sectional area are known. |
| \(g\) | Acceleration due to gravity. | Use 9.80665 m/s² in SI units or 32.174 ft/s² in US customary units. |
| \(D\) | Hydraulic depth or characteristic length. | For open channels, use \(D=A/T\). For model scaling, use the defined characteristic length. |
| \(Q\) | Discharge or volumetric flow rate. | Use the flow rate through the section being evaluated. |
| \(A\) | Flow area. | Use the wetted cross-sectional area of flowing water, not total channel area above the water surface. |
| \(T\) | Top width of the water surface. | Use the free-surface width, especially for nonrectangular channels. |
| \(V_c\) | Critical velocity. | Calculated as \(V_c=\sqrt{gD}\), where \(Fr=1\). |
How to Use the Calculator
Start with the values you know. For a simple problem, enter velocity and hydraulic depth. For an open-channel geometry problem, enter discharge, area, and top width so the calculator can compute \(V\), \(D\), and \(Fr\).
Choose what you want to solve for
Select Froude number, velocity, hydraulic depth, gravity, or critical velocity depending on the known values.
Select the correct input method
Use simple mode for known velocity and depth. Use open-channel geometry mode when you know \(Q\), \(A\), and \(T\).
Check hydraulic depth carefully
Use \(D=A/T\), not hydraulic radius. Hydraulic radius \(R=A/P\) belongs in many resistance equations, but it is not the usual Froude number depth term.
Review the flow classification
Use the result to classify the flow as subcritical, near critical, or supercritical, then check whether the classification makes sense for the channel.
How to Interpret the Froude Number Result
A Froude number below 1 usually means slow, deep, subcritical flow. A value near 1 means critical flow, and a value above 1 means fast, shallow, supercritical flow.
| Froude Number | Flow Regime | What It Usually Means | What to Check Next |
|---|---|---|---|
| \(Fr<1\) | Subcritical | Slower, deeper flow. Downstream conditions can influence upstream water depth. | Check tailwater, backwater effects, and channel control. |
| \(Fr\approx1\) | Critical or near critical | Velocity is close to gravity wave speed. Small changes can shift the regime. | Check depth sensitivity, critical depth, and energy conditions. |
| \(Fr>1\) | Supercritical | Faster, shallower flow. Disturbances generally move downstream. | Check hydraulic jump potential, erosion risk, and downstream transition. |
| Very high \(Fr\) | Strongly supercritical | The input may represent steep, shallow, high-velocity flow or a unit/depth error. | Verify depth units, velocity, and whether the section is physically realistic. |
| Negative or impossible | Invalid | Negative depth, negative gravity, or inconsistent inputs are not physically valid. | Recheck signs, units, and geometry values. |
What to do with the result
Use the result as a flow-regime flag. Subcritical flow often requires attention to downstream control and backwater effects. Supercritical flow often requires attention to hydraulic jumps, erosion, transitions, and energy dissipation. Near-critical results deserve extra review because small field or rounding errors can change the classification.
What changes the result most?
Velocity has a direct effect on \(Fr\), while hydraulic depth affects the denominator through a square root. This means doubling velocity approximately doubles Froude number, but doubling hydraulic depth reduces Froude number by about \(1/\sqrt{2}\). Shallow-depth errors can therefore make supercritical results look much larger than expected.
Quick sanity check
In a rectangular channel, hydraulic depth equals flow depth, so \(D=y\). In a nonrectangular or irregular channel, use \(D=A/T\). If your result changes dramatically when switching from depth to hydraulic radius, you probably used the wrong depth term.
Input Quality Checklist
Froude number calculations are simple, but the input definitions are easy to mix up. Check these items before relying on the result.
Use Hydraulic Depth
For open-channel flow, use \(D=A/T\), not hydraulic radius \(R=A/P\).
Match Unit Systems
Use m/s with m and m/s², or ft/s with ft and ft/s². Do not mix SI and US customary units.
Use Mean Velocity
For channel sections, use average velocity \(V=Q/A\), not a localized point velocity unless that is the intended analysis.
Check Top Width
For irregular channels, top width \(T\) should represent the water-surface width at the flow depth being analyzed.
Step-by-Step Worked Example
This example calculates Froude number for an open-channel section using discharge, flow area, and top width. That is a common engineering workflow because measured or modeled channel sections often provide \(Q\), \(A\), and \(T\) rather than hydraulic depth directly.
Calculate Velocity
Calculate Hydraulic Depth
Calculate Froude Number
Result
Froude number: \(Fr\approx0.74\), so the flow is subcritical.
Is this result reasonable?
Yes. A Froude number below 1 is reasonable for relatively deeper, slower open-channel flow. Because this section is subcritical, downstream conditions may influence the upstream water surface.
Froude Number Flow Diagram
A good Froude number diagram should show the difference between subcritical, critical, and supercritical flow. It should also show that hydraulic depth is tied to the channel cross section, not the wetted perimeter.
Typical Froude Number Values and Reference Ranges
Froude number ranges are interpretation ranges, not universal design limits. The same value can mean different design concerns in rivers, flumes, spillways, culverts, and laboratory models.
| Range | Common Interpretation | Practical Note |
|---|---|---|
| \(Fr<0.5\) | Clearly subcritical for many open-channel checks. | Often less sensitive to small depth changes than near-critical flow. |
| \(0.5\le Fr<0.95\) | Subcritical, but moving toward critical behavior. | Check downstream control and whether velocity is increasing through transitions. |
| \(0.95\le Fr\le1.05\) | Near critical. | Treat carefully; rounding and field measurement error can change the classification. |
| \(1.05<Fr\le3\) | Supercritical. | Check hydraulic jump potential, transitions, and erosion protection. |
| \(Fr>3\) | Strongly supercritical. | Verify the inputs and review energy dissipation, air entrainment, and downstream controls. |
Design Ranges and Practical Checks
A mathematically correct Froude number does not automatically mean a channel design is acceptable. It only identifies the balance between inertia and gravity effects at the selected section.
Subcritical Checks
Review tailwater, backwater, downstream control, and whether the water surface profile is controlled by downstream conditions.
Near-Critical Checks
Small changes in flow depth, slope, or geometry can change the regime, so near-critical results deserve conservative review.
Supercritical Checks
Review erosion, hydraulic jumps, channel transitions, stilling basins, and downstream depth requirements.
Engineering judgment check
For final design, do not evaluate Froude number at only one location if the channel geometry, slope, or flow rate changes. Check the regime upstream and downstream of transitions, structures, contractions, expansions, grade breaks, and potential hydraulic jump locations.
Unit Conversion Notes
Froude number is dimensionless, but the inputs must still use a consistent unit system. Mixing SI and US customary units is one of the fastest ways to get a believable but wrong answer.
| Quantity | Common Units | Conversion Reminder |
|---|---|---|
| Velocity | m/s, ft/s, mph, km/h | \(1\,ft/s=0.3048\,m/s\), \(1\,mph=0.44704\,m/s\) |
| Length or Hydraulic Depth | m, ft, cm, in | \(1\,ft=0.3048\,m\), \(1\,in=0.0254\,m\) |
| Gravity | m/s², ft/s² | Use \(9.80665\,m/s^2\) or \(32.174\,ft/s^2\) |
| Discharge | m³/s, ft³/s, L/s, gpm | \(1\,ft^3/s=0.0283168\,m^3/s\) |
| Area | m², ft² | \(1\,ft^2=0.092903\,m^2\) |
Unit-system rule
If velocity is in ft/s, use \(g\) in ft/s² and depth in ft. If velocity is in m/s, use \(g\) in m/s² and depth in m. The final Froude number has no unit, but the input consistency controls the answer.
Froude Number vs. Related Flow Calculations
Froude number is a flow-regime and wave-speed parameter. It should not be confused with Manning’s equation, Reynolds number, or hydraulic radius calculations, even though these concepts often appear in the same open-channel workflow.
| Calculation | Main Purpose | Key Input Trap | Use It When |
|---|---|---|---|
| Froude Number | Classifies subcritical, critical, or supercritical flow. | Uses hydraulic depth \(D=A/T\), not hydraulic radius. | You need flow-regime interpretation. |
| Manning’s Equation | Estimates open-channel velocity, discharge, or slope. | Uses hydraulic radius \(R=A/P\), roughness, and slope. | You need open-channel flow capacity or normal depth checks. |
| Reynolds Number | Compares inertial and viscous effects. | Depends on viscosity and characteristic length. | You need laminar, transitional, or turbulent flow context. |
| Bernoulli Equation | Relates pressure head, velocity head, elevation head, and losses. | Requires careful assumptions about losses and open-channel pressure head. | You need energy balance or head calculations. |
Common Mistakes That Cause Wrong Results
Most wrong Froude number results come from using the wrong depth term, mixing units, or applying a section average formula to a poorly defined cross section.
Common Mistakes
- Using hydraulic radius \(R=A/P\) instead of hydraulic depth \(D=A/T\).
- Mixing m/s velocity with ft/s² gravity or feet of depth.
- Using total channel area instead of wetted flow area.
- Using a local velocity measurement when the formula needs mean section velocity.
- Ignoring near-critical sensitivity when \(Fr\) is close to 1.
Better Practice
- Calculate hydraulic depth from \(D=A/T\) for nonrectangular channels.
- Keep velocity, gravity, and depth in one consistent unit system.
- Use \(V=Q/A\) when discharge and area are known.
- Check upstream and downstream sections when geometry changes.
- Treat values from about 0.95 to 1.05 as near critical rather than forcing a hard classification.
Troubleshooting Unexpected Results
If the calculator result looks unrealistic, start by checking depth definition and unit consistency before changing the formula.
| Problem | Likely Cause | Fix |
|---|---|---|
| Froude number is much higher than expected | Depth may be too small, velocity may be too high, or units may be mixed. | Check depth units, velocity units, and whether \(D=A/T\) was calculated correctly. |
| Froude number is extremely low | Depth may be too large or velocity may have been entered in the wrong unit. | Verify whether velocity is in m/s, ft/s, mph, or km/h. |
| Result changes when using Manning’s hydraulic radius | Hydraulic radius was substituted for hydraulic depth. | Use hydraulic depth \(D=A/T\) for the Froude number formula. |
| Flow classification is close to critical | The result is sensitive to small changes in depth, velocity, or geometry. | Review input precision and consider a near-critical range instead of a hard cutoff. |
| Geometry result looks wrong | Area or top width may not represent the same water depth and cross section. | Use area and top width from the same section and flow condition. |
Common edge cases
Very shallow flow, rapidly varied flow, hydraulic jumps, bends, contractions, expansions, and irregular field sections can make a single-section Froude number misleading. The value may still be mathematically correct, but it may not describe the entire hydraulic behavior of the site.
Assumptions, Sources, and Limitations
This calculator is intended for education, quick checks, and preliminary engineering review. It uses standard steady open-channel relationships and assumes the entered section values represent the same flow condition.
Formula Assumption
The main formula assumes a representative mean velocity and a hydraulic depth or characteristic length appropriate for the section.
Geometry Assumption
Area \(A\), top width \(T\), and discharge \(Q\) must describe the same channel section and water surface.
Application Limit
The calculator does not model full water-surface profiles, sediment transport, hydraulic jumps, roughness losses, or tailwater controls.
Final Design Note
For final hydraulic design, verify results with project criteria, field data, channel geometry, energy calculations, erosion checks, and professional engineering judgment.
Calculation basis
The calculation is based on the standard open-channel definition of Froude number as a dimensionless ratio of inertial effects to gravity effects. For additional open-channel context, see Iowa State University’s open-channel flow reference: Open Channel Flow.
Glossary of Terms
These terms help explain the calculator result and the most common open-channel flow interpretation issues.
Froude Number
A dimensionless value comparing flow velocity to gravity wave speed.
Subcritical Flow
Flow with \(Fr<1\), usually slower and deeper, where downstream conditions can influence upstream water depth.
Critical Flow
Flow near \(Fr=1\), where velocity is close to gravity wave speed and the flow is highly sensitive to changes.
Supercritical Flow
Flow with \(Fr>1\), usually faster and shallower, where disturbances generally move downstream.
Hydraulic Depth
The open-channel depth term \(D=A/T\), equal to flow area divided by top width.
Hydraulic Radius
The geometry term \(R=A/P\), equal to flow area divided by wetted perimeter. It is not the usual Froude number depth term.
Frequently Asked Questions
What does the Froude Number Calculator calculate?
The Froude Number Calculator calculates the dimensionless Froude number and classifies flow as subcritical, near critical, or supercritical using velocity, gravity, and hydraulic depth or characteristic length.
What is the Froude number formula?
The common open-channel formula is \(Fr=V/\sqrt{gD}\), where \(V\) is mean flow velocity, \(g\) is gravitational acceleration, and \(D\) is hydraulic depth or characteristic length.
Is hydraulic depth the same as hydraulic radius?
No. Hydraulic depth is \(D=A/T\), where \(A\) is flow area and \(T\) is top width. Hydraulic radius is \(R=A/P\), where \(P\) is wetted perimeter. Froude number normally uses hydraulic depth, not hydraulic radius.
What does a Froude number greater than 1 mean?
A Froude number greater than 1 indicates supercritical flow, which is typically fast and shallow. Disturbances generally move downstream, and a transition to subcritical flow can form a hydraulic jump.
Why does my Froude number result look wrong?
The most common causes are mixed units, entering hydraulic radius instead of hydraulic depth, using a local velocity instead of mean velocity, or using area and top width from different flow conditions.
Can I use the Froude number result for final hydraulic design?
Use the calculator for education, preliminary checks, and quick flow-regime classification. Final hydraulic design should also verify channel geometry, flow data, energy grade line, hydraulic jumps, erosion risk, field conditions, and applicable project requirements.