Froude number (Fr)
The Froude number (Fr) is a dimensionless number that relates inertial forces and the local force of gravity.
*Simplified calculation of local gravity.
The Froude number is a fundamental dimensionless parameter in hydraulic studies, used to understand free surface (non-pressurized) flows. It is defined as the ratio of flow velocity to the propagation speed of gravity waves at the surface, or as the ratio of inertial forces to the local force of gravity. Its mathematical expression is:
Where:
- Fr: dimensionless Froude number.
- v: average axial velocity of the fluid.
- g: local gravity.
- L: hydraulic depth of the channel.
This number is used to determine whether the flow regime is subcritical, critical, or supercritical.
Subcritical regime (Fr < 1): Characterized by low velocities and high depths, where gravity dominates the flow. Disturbances can propagate both upstream and downstream.
Critical regime (Fr = 1): At this point, inertial and gravitational forces are balanced. The flow reaches the minimum specific energy for a given flow rate, and any disturbance can cause a transition to a regime. This phenomenon is precisely what is used for flow measurement in regime-changing gauges such as Parshall flumes or Long Throat Gauges (LTGs).
Supercritical regime (Fr > 1) In this regime, the flow exhibits high velocities and low depths (normal behavior with increasing velocity), dominated by inertia. Disturbances only propagate downstream, which means that flow control must be achieved from upstream.
When it is necessary to know the regime that exists in a channel or river without the need to take measurements, there is an empirical method used by hydraulic experts. This is important since to perform the gauging of a river, a Froude number < 0.6 is required, just as if a regime change gauging system is installed, it is required that before the control zone, a subcritical regime is had precisely so that when passing through the Parshall, AGL, etc., a regime change occurs to supercritical and the height at the critical depth can be measured and the flow circulating in that river can be known.
Therefore, the method known as putting the stick in consists of putting a stick into the river and checking the behavior of the waves that are generated as in the following image:
With enough experience, you will be able to empirically determine the approximate Froude number without having a level and speed meter at hand.
Worked example
Rectangular channel with mean velocity v = 1.20 m/s, flow depth y = 0.50 m and g = 9.80665 m/s²:
Fr = 1.20 / √(9.80665 × 0.50) = 1.20 / 2.214 = 0.54
Since Fr < 1, the flow is subcritical. If the same channel carries v = 3.00 m/s at a depth of 0.25 m, √(9.80665 × 0.25) = 1.566 m/s and Fr = 3.00 / 1.566 = 1.92: the flow is supercritical.
Non-rectangular channels: hydraulic depth
In trapezoidal or circular channels, L is not the flow depth but the hydraulic depth D = A / T (wetted area divided by top width). Example: trapezoidal channel with bottom width b = 2.0 m, side slope z = 1.5 and depth y = 0.80 m carrying Q = 3.0 m³/s:
A = (b + z·y)·y = 2.56 m²
T = b + 2·z·y = 4.40 m
D = A / T = 0.582 m
v = Q / A = 1.172 m/s
Fr = 1.172 / √(9.80665 × 0.582) = 0.49 (subcritical)
Hydraulic jump type by upstream Froude number
Fr 1.0 to 1.7: undular jump
Fr 1.7 to 2.5: weak jump
Fr 2.5 to 4.5: oscillating jump
Fr 4.5 to 9.0: steady jump, best energy dissipation
Fr above 9.0: strong jump
Frequently asked questions
Which value of g should I use?
9.80665 m/s² is standard gravity. Local gravity ranges from about 9.78 to 9.83 m/s² depending on latitude and altitude (you can compute it with the local gravity calculator on this page). In the first example, using 9.78 changes Fr from 0.542 to 0.543, so the effect is small.
What is critical depth?
It is the depth at which Fr = 1. In a rectangular channel it equals yc = (q² / g)^(1/3), where q is the flow rate per metre of width. In the first example, q = 0.60 m²/s and yc = 0.33 m; since the actual depth (0.50 m) is larger, the flow is subcritical.
Does the Froude number depend on viscosity?
No. Viscous effects are described by the Reynolds number; the Froude number compares inertia with gravity and governs free-surface flows.
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