Froude number

# Froude number

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Encyclopedia
The Froude number is a dimensionless number defined as the ratio of a characteristic velocity to a gravitational wave velocity. It may equivalently be defined as the ratio of a body's inertia to gravitational forces. In fluid mechanics
Fluid mechanics
Fluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...

, the Froude number is used to determine the resistance of an object moving through water, and permits the comparison of objects of different sizes. Named after William Froude
William Froude
William Froude was an English engineer, hydrodynamicist and naval architect. He was the first to formulate reliable laws for the resistance that water offers to ships and for predicting their stability....

, the Froude number is based on the speed/length ratio as defined by him.

The Froude number is defined as:
where is a characteristic velocity , and is a characteristic water wave propagation velocity. The Froude number is thus analogous to the Mach number
Mach number
Mach number is the speed of an object moving through air, or any other fluid substance, divided by the speed of sound as it is in that substance for its particular physical conditions, including those of temperature and pressure...

. The greater the Froude number, the greater the resistance.

## Origins

In open channel flows, Bélanger (1828) introduced first the ratio of the flow velocity to the square root of the gravity acceleration times the flow depth. When the ratio was less than unity, the flow behaved like a fluvial motion (i.e. subcritical flow), and like a torrential flow motion when the ratio was greater than unity (Chanson 2009).

Quantifying resistance of floating objects is generally credited to William Froude
William Froude
William Froude was an English engineer, hydrodynamicist and naval architect. He was the first to formulate reliable laws for the resistance that water offers to ships and for predicting their stability....

, who used a series of scale models to measure the resistance each model offered when towed at a given speed. Froude's observations led him to derive the Wave-Line Theory which first described the resistance of a shape as being a function of the waves caused by varying pressures around the hull as it moves through the water. The naval constructor Ferdinand Reech had put forward the concept in 1832 but had not demonstrated how it could be applied to practical problems in ship resistance. Speed/length ratio was originally defined by Froude in his Law of Comparison in 1868 in dimensional terms as:

where:
v = speed in knots
LWL = length of waterline in feet

The term was converted into non-dimensional terms and was given Froude's name in recognition of the work he did. In France, it is sometimes called Reech–Froude number after Ferdinand Reech.

### Ship hydrodynamics

For a ship, the Froude number is defined as:

where V is the velocity of the ship, g is the acceleration due to gravity
Gravitational constant
The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...

, and L is the length of the ship at the water line level, or Lwl in some notations. It is an important parameter with respect to the ship's drag
Drag (physics)
In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...

, or resistance, including the wave making resistance
Wave making resistance
Wave making resistance is a form of drag that affects surface watercraft, such as boats and ships, and reflects the energy required to push the water out of the way of the hull. This energy goes into creating the wake.-Physics:...

. Note that the Froude number used for ships, by convention, is the square root of the Froude number as defined above.

### Shallow water waves

For shallow water waves, like for instance tidal waves
Tsunami
A tsunami is a series of water waves caused by the displacement of a large volume of a body of water, typically an ocean or a large lake...

and the hydraulic jump
Hydraulic jump
A hydraulic jump is a phenomenon in the science of hydraulics which is frequently observed in open channel flow such as rivers and spillways. When liquid at high velocity discharges into a zone of lower velocity, a rather abrupt rise occurs in the liquid surface...

, the characteristic velocity V is the average
Average
In mathematics, an average, or central tendency of a data set is a measure of the "middle" value of the data set. Average is one form of central tendency. Not all central tendencies should be considered definitions of average....

flow velocity, averaged over the cross-section perpendicular to the flow direction. The wave velocity, c, is equal to the square root of gravitational acceleration g, times cross-sectional area A, divided by free-surface width B:
so the Froude number in shallow water is:
For rectangular cross-sections with uniform depth d, the Froude number can be simplified to:
For Fr < 1 the flow is called a subcritical flow, further for Fr > 1 the flow is characterised as supercritical flow
Supercritical flow
A supercritical flow is when the flow velocity is larger than the wave velocity. The analogous condition in gas dynamics is supersonic....

. When Fr ≈ 1 the flow is denoted as critical flow.

An alternate definition used in fluid mechanics is
where each of the terms on the right have been squared. This form is the reciprocal of the Richardson number.

## Extended Froude number

Geophysical mass flows such as avalanches and debris flows take place on inclined slopes which then merges into a gentle and flat run-out zones.

So, these flows are associated with the elevation of the topographic slopes that induce the gravity potential energy together with the pressure potential energy during the flow. Therefore, the classical Froude number should include this additional effect. For such a situation, Froude number needs to be re-defined. The extended Froude number is defined as the ratio between the kinetic and the potential energy:
where is the mean flow velocity, , ( is the earth pressure coefficient, is the slope), , is the channel downslope position and is the distance from the point of the mass release along the channel to the point where the flow hits the horizontal reference datum; and are the pressure potential and gravity potential energies, respectively. In the classical definition of the shallow-water or granular flow Froude number, the potential energy associated with the surface elevation, , is not considered. The extended Froude number differs substantially from the classical Froude number for higher surface elevations. The term emerges from the change of the geometry of the moving mass along the slope. Dimensional analysis suggests that for shallow flows is of order , while and are both of order unity. If the mass is shallow with a virtually bed-parallel free-surface, then can be disregarded. In this situation, if the gravity potential is not taken into account, then Fr is unbounded even though the kinetic energy is bounded. So, formally considering the additional contribution due to the gravitational potential energy, the singularity in Fr is removed.

### Stirred tanks

In the study of stirred tanks, the Froude number governs the formation of surface vortices. Since the impeller tip velocity is proportional to , where is the impeller speed (rev/s) and is the impeller diameter, the Froude number then takes the following form:

### Densimetric Froude number

When used in the context of the Boussinesq approximation the densimetric Froude number is defined as
where is the reduced gravity:

The densimetric Froude number is usually preferred by modellers who wish to nondimensionalize a speed preference to the Richardson number which is more commonly encountered when considering stratified shear layers. For example, the leading edge of a gravity current
Gravity current
In fluid dynamics, a gravity current is a primarily horizontal flow in a gravitational field that is driven by a density difference, hence gravity currents also sometimes being referred to as "density currents"...

moves with a front Froude number of about unity.

### Walking Froude number

The Froude number may be used to study trends in animal gait patterns. In analyses of the dynamics of legged locomotion, a walking limb is often modeled as an inverted pendulum
Pendulum
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced from its resting equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position...

, where the center of mass goes through a circular arc centered at the foot. The Froude number is the ratio of the centripetal force around the center of motion, the foot, and the weight of the animal walking:

where is the mass, is the characteristic length, is the acceleration due to gravity
Earth's gravity
The gravity of Earth, denoted g, refers to the acceleration that the Earth imparts to objects on or near its surface. In SI units this acceleration is measured in metres per second per second or equivalently in newtons per kilogram...

and is the velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

. The characteristic length, , may be chosen to suit the study at hand. For instance, some studies have used the vertical distance of the hip joint from the ground , while others have used total leg length .

The Froude number may also be calculated from the stride frequency as follows:

If total leg length is used as the characteristic length, then the theoretical maximum speed of walking has a Froude number of since any higher value would result in 'take-off' and the foot missing the ground. The typical transition speed from bipedal running
Running
Running is a means of terrestrial locomotion allowing humans and other animals to move rapidly on foot. It is simply defined in athletics terms as a gait in which at regular points during the running cycle both feet are off the ground...

to walking occurs with . R. McN. Alexander found that animals of different sizes and masses travelling at different speeds, but with the same Froude number, consistently exhibit similar gaits. This study found that animals typically switch from an amble to a symmetric running gait (e.g. a trot or pace) around a Froude number of . A preference for asymmetric gaits (e.g. a canter, transverse gallop, rotary gallop, bound, or pronk) was observed at Froude numbers between and .

## Uses

The Froude number is used to compare the wave making resistance
Wave making resistance
Wave making resistance is a form of drag that affects surface watercraft, such as boats and ships, and reflects the energy required to push the water out of the way of the hull. This energy goes into creating the wake.-Physics:...

between bodies of various sizes and shapes.

In free-surface flow, the nature of the flow (supercritical
Supercritical flow
A supercritical flow is when the flow velocity is larger than the wave velocity. The analogous condition in gas dynamics is supersonic....

or subcritical) depends upon whether the Froude number is greater than or less than unity.

The Froude number has been used to study trends in animal locomotion in order to better understand why animals use different gait patterns as well as to form hypotheses about the gaits of extinct species .