What This Solves
Calculates the Froude number for open channel flow and classifies the flow regime as subcritical (Fr < 1), critical (Fr = 1), or supercritical (Fr > 1).
Best Used When
- You need to classify whether channel flow is subcritical or supercritical
- You are checking for the potential of a hydraulic jump at a slope transition or structure
- You want a quick check of flow regime from known depth and velocity
Do NOT Use When
- You need the actual critical depth value rather than just the Froude number — Use Critical Depth Calculator
- You need the normal (uniform) flow depth for a given discharge — Use Normal Depth Calculator
Key Assumptions
- Froude number is calculated as Fr = V / sqrt(g * D_h) where D_h is hydraulic depth
- Hydrostatic pressure distribution exists at the cross-section
- Velocity is uniform across the cross-section (or average velocity is representative)
- The channel has a well-defined cross-section geometry
Input Quality Notes
For non-rectangular channels, the hydraulic depth (A/T) is used rather than the flow depth. Ensure you are using the correct depth definition for your channel shape.
Calculate the Froude number for open channel flow and classify the regime as subcritical, critical, or supercritical. Enter velocity and hydraulic depth directly, or supply a channel shape with depth and either velocity or discharge to let the tool derive the geometry.
Froude Number Overview
The Froude number is a dimensionless parameter representing the ratio of inertial forces to gravitational forces in open channel flow:
Fr = V / sqrt(g * Dh)
where:
- V = flow velocity
- g = gravitational acceleration
- Dh = hydraulic depth = A / T
Flow Regime Classification
| Regime | Fr Range | Characteristics |
|---|---|---|
| Subcritical | Fr < 1 | Deep, slow flow; downstream control |
| Critical | Fr = 1 | Unstable transition; minimum energy |
| Supercritical | Fr > 1 | Shallow, fast flow; upstream control |
Source: Chow (1959), Open-Channel Hydraulics.
For educational purposes only. Not a substitute for professional engineering judgment.
How the Froude number is calculated
The Froude number compares the inertial forces driving the flow to the gravitational forces restoring the free surface. It is computed from the flow velocity and the hydraulic depth:
Fr = V / √(g · Dh)
where:
- Fr = Froude number (dimensionless)
- V = mean flow velocity (ft/s or m/s). If discharge Q is entered instead, V = Q / A.
- g = gravitational acceleration = 32.174 ft/s² (US customary) or 9.81 m/s² (SI)
- Dh = hydraulic depth = A / T, the flow area divided by the free-surface top width
The denominator is the wave celerity, c = √(g · Dh), the speed of a shallow-water surface wave. The Froude number is therefore the ratio of flow speed to wave speed, Fr = V / c. The calculator also reports the specific energy, E = y + V²/(2g), which reaches its minimum at critical flow.
Flow is then classified by the value of Fr:
- Fr < 1 → subcritical (tranquil) flow — deep, slow, downstream control
- Fr = 1 → critical flow — minimum specific energy, hydraulic control point
- Fr > 1 → supercritical (rapid) flow — shallow, fast, upstream control
Flow within the band 0.9 < Fr < 1.1 is flagged as near-critical, an unstable zone that is generally avoided in design. Source: Chow (1959), Open-Channel Hydraulics, Chapter 3; FHWA HEC-22.
Hydraulic depth by channel shape
The Froude number depends on the hydraulic depth Dh = A/T, which differs from the flow depth y for every shape except rectangular. The calculator derives these from the geometry you enter; the table below summarizes the relationships it uses.
| Shape | Flow area, A | Top width, T | Hydraulic depth, Dh = A/T |
|---|---|---|---|
| Rectangular | b · y | b | y |
| Triangular (V-shape) | z · y² | 2 z y | y / 2 |
| Trapezoidal | (b + z y) y | b + 2 z y | (b + z y) y / (b + 2 z y) |
| Circular (part-full) | from depth ratio y/D | chord at water surface | A / T (geometry-based) |
| Circular (just full) | π D² / 4 | — | D / 4 (convention) |
Symbols: b = bottom width, y = flow depth, z = side slope (horizontal:vertical, z:1), D = pipe diameter. Source: Chow (1959), Open-Channel Hydraulics.
Worked example
A 10 ft wide rectangular channel flows 3 ft deep at a mean velocity of 3 ft/s (US customary, g = 32.174 ft/s²):
- For a rectangular channel, Dh = y = 3 ft
- Wave celerity c = √(32.174 × 3) = 9.83 ft/s
- Froude number Fr = 3 / 9.83 = 0.305
- Because Fr < 1, the flow is subcritical (tranquil)
Raising the velocity at the same 2 ft depth to about 8.0 ft/s would give Fr ≈ 1.0 (critical), and 10 ft/s at 1 ft depth gives Fr ≈ 1.76 (supercritical) — the three regimes from the library's verification cases.
Frequently asked questions
What is the Froude number in open channel flow?
The Froude number (Fr) is a dimensionless ratio of inertial forces to gravitational forces in open channel flow, defined as Fr = V / sqrt(g · D_h), where V is the mean velocity, g is gravitational acceleration (32.174 ft/s² or 9.81 m/s²) and D_h is the hydraulic depth (flow area divided by top width, A/T). It is the single number that classifies the flow regime as subcritical, critical, or supercritical.
What is the difference between subcritical and supercritical flow?
Subcritical flow (Fr < 1) is deep and slow ("tranquil") — disturbances can travel upstream, so the water surface is controlled by downstream conditions and backwater effects propagate upstream. Supercritical flow (Fr > 1) is shallow and fast ("rapid") — disturbances only travel downstream, the flow is controlled by upstream conditions, and high velocities can cause erosion. At Fr = 1 the flow is critical, sitting at minimum specific energy for the given discharge.
Why should critical flow (Fr = 1) be avoided in design?
At critical flow the specific energy is at its minimum for the discharge, so the depth-energy curve is nearly vertical: small changes in bed slope, roughness, or discharge produce large, unstable changes in depth. The water surface tends to oscillate and form standing waves. Channels are normally designed to keep the Froude number clear of the near-critical band (roughly 0.9 < Fr < 1.1), reserving Fr = 1 only for deliberate control sections such as flumes and weir crests.
How is hydraulic depth different from flow depth?
Hydraulic depth D_h = A/T is the flow area divided by the free-surface (top) width, and it is the length scale used in the Froude number — not the maximum flow depth y. For a rectangular channel the two are equal (D_h = y). For a triangular (V-shaped) channel D_h = y/2, and for trapezoidal or circular sections D_h is computed from the actual geometry. Using flow depth instead of hydraulic depth in non-rectangular sections will give the wrong Froude number.
What is wave celerity and how does it relate to the Froude number?
Wave celerity is the speed of a small gravity (shallow-water) surface wave, c = sqrt(g · D_h). The Froude number is simply the ratio of flow velocity to wave celerity, Fr = V / c. When Fr < 1 the flow moves slower than a wave, so waves can travel both upstream and downstream; when Fr > 1 the flow outruns its own waves and disturbances cannot move upstream.
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Last verified: February 2026