Froude number

Physical key figure
Surname	Froude number
Formula symbol	${\displaystyle {\mathit {Fr}}}$
dimension	dimensionless
definition	${\displaystyle {\mathit {Fr}}={\frac {v}{\sqrt {gL}}}{\text{ oder }}{\mathit {Fr}}^{\prime }={\frac {v^{2}}{gL}}}$
${\displaystyle v}$ Flow velocity ${\displaystyle g}$ Gravity acceleration ${\displaystyle L}$ characteristic length
Named after	William Froude
scope of application	Currents with a free surface

The Froude number ( symbol :  Fr ) is a dimensionless number in physics . It is named after William Froude  (1810–1879) and represents a measure of the ratio of inertial forces to gravitational forces within a hydrodynamic system. It plays a role , for example, in hydrodynamics with the influence of the free liquid surface and is used to describe flows in open areas Flushing or used by bow waves of ships. The Froude number is, along with the Reynolds number, one of the coefficients of the dimensionless Navier-Stokes equation .

definition

For historical reasons, two definitions are used for the Froude number.

${\displaystyle {\mathit {Fr}}={\frac {v}{\sqrt {gL}}}}$   or their square    ${\displaystyle {\mathit {Fr}}^{\prime }={\frac {v^{2}}{gL}}}$

each with

• a characteristic flow velocity  v
• a characteristic length  L
• the acceleration due to gravity  g .

The same physical relationship is behind both definitions. When applying the Froude number, it is important to note which definition was used.

Froude number for open channels

If the water depth of an open channel is used for the characteristic length  L , the Froude number describes the ratio of flow velocity and the propagation velocity of a shallow water wave :

${\displaystyle Fr={\frac {v_{fl}}{v_{ausbr}}}}$

This characterizes the flow condition of an open channel:

• Static flow condition ( ): A disturbance (e.g. a wave that occurs when a stone is thrown into the water) spreads evenly in all directions, i.e. in a circle. The descriptive differential equation is called elliptical (special case of the flowing state). Example: lake . ${\displaystyle Fr=0\Leftrightarrow v_{fl}=0}$
• Current state of flow ( ): Faults propagate both upstream and downstream. The wave propagation shows a parabolic pattern. The flow is described by a parabolic differential equation. Example: river . ${\displaystyle Fr<1\Leftrightarrow v_{fl}<v_{ausbr}}$
• Limit discharge / critical discharge ( ): Waves can no longer propagate against the current. The wave front directed towards the upper current “stops” at the point of disturbance (analogous to the sound barrier ). In this state, the greatest possible amount of water can be drained off at the present energy level. In hydraulic engineering , this is used as a flow control. Example: overflowing a weir . ${\displaystyle Fr=1\Leftrightarrow v_{fl}=v_{ausbr}}$
• Shooting flow condition ( ): A disturbance now only propagates downstream. Propagation patterns and the associated differential equation are called hyperbolic , example: mountain stream . ${\displaystyle Fr>1\Leftrightarrow v_{fl}>v_{ausbr}}$

Relationships for ship model tests

If forces due to viscosity have only a minor influence, the behavior of a ship on the surface of the liquid can be shown in a model experiment . In order for such investigations on the ship model to result in comparable flow conditions with regard to the waves as in the original, the Froude number of the original and the model must match. This is the case when the ratio of the length to the square of the speed is identical. The different measured variables can then be converted as follows:

• Lengths with the length scale
• Times with the square root of the length scale
• Forces to the third power of the length scale ( assuming the same density of the fluid)
• Accelerations are the same in model and large version.

Web links

• Froude number in the lexicon of geosciences
• Heinz Herwig: Fluid Mechanics A – Z: A systematic classification of terms and concepts in fluid mechanics

See also

• Dimensional analysis
• Shallow water effect

literature

• Jürgen Zierep: Similarity laws and model rules in fluid mechanics . Karlsruhe 1991, ISBN 3-7650-2041-9 .