Froude Number Formula: Definition, Significance

Froude number is a dimensionless quantity that elucidates the effect of gravity on fluid motion. Froude number comes into the picture when concerned about fluid mechanics and hydrology. Froude number is denoted by ‘Fr’ and is applicable in many fluid models. The dimensionless numbers are generally used to analyze the prototype models. And these numbers are Reynolds numbers, Weber’s number, and Froude’s number. The Froude number governs dynamic similarity in such situations where the gravitational force is important. For instance, flow through the open channels, flow over the spillway of the dam, etc. Let’s learn more about the Froude number formula and discuss some of the important questions.

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Definition of Froude Number 

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Froude number is a dimensionless quantity that was named after scientist William Froude. Froude number is known as the flow of inertia to the external felid. Froude number as described by William Froude is based on speed length ratio. In fluid mechanics, the Froude number helps to know where inertia is greater than the effects of gravity in the fluid medium. A falling jet stream is an example of a Froude number where a jet cylinder is accelerated under the presence of gravity.

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What is Dimensionless Quantity?

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Dimensionless quantity is a quantity that is without any physical dimension. Dimensionless quantities are described as the ratio of quantities that are not dimensionless means ratio of quantities with dimensions. But further, they get multiplied thus canceling their dimensions and resulting in being dimensionless.

Dimensionless quantities come in handy in comparing quantities or systems which are of different types and means. Such as for Froude number, Reynolds number, and Weber number.

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Froude Number Formula

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Froude number in fluid mechanics as per definition is the ratio of inertia to the gravitational force. It is a dimensionless quantity is used to show the influence of gravity on the motion of the fluid. Froude number in formula is defined as –

\(Fr = \frac{V}{\sqrt{gD}}\)

Where -

• Fr is Froude Number 

• V is Velocity 
• g is Gravity 
• D is Hydraulic Depth or depth of flow (I)

Froude number can also be represented as – 

\(F_r = (\frac{inertia force}{gravity force})^{1/2}\)

Where we know inertia is -

\(Inertia force F_r = p l^2V^2\)

And, Gravity force = mass * gravitational acceleration 

\(= pl^3g\)

In other words, the Froude number formula is described as Fr = v/(gD)^½. In this Fr is Froude number, V is the velocity, g is gravitational force and D is the depth of flow or hydraulic flow. In this Fr is equaled to root to the ratio of velocity to the gravity force.

Various conditions are concerned with the Froude number. When the Froude number is smaller than 1, small surface waves move upwards. When the Froude number is greater than 1, small stream waves move downwards. And when the Froude number is equaled to 1 is known as the Critical Froude number. When it is equaled to 1, the surface, velocity gets equaled to the velocity of flow.

In short when, 

• Froude number, Fr < 1 – Surface waves moves upward 
• Froude number, Fr > 1 – Surface wanes moves downward 
• Froude number, Fr = 1 – Surface wave velocity is equaled to the velocity of flow (Critical Froude Number).

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Significance of Froude Number 

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Being a dimensionless quantity Froude number is used in the comparison. It measures the comparison of waves which makes resistance between bodies of various sizes and shapes.

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Difference Between Froude number and Reynolds Number

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Froude number and Reynolds number both come under the umbrella of dimensionless numbers. As being dimensional numbers, both are used in the comparison of quantities of various bodies. Reynolds number is a dimensionless number that is known to determine the pattern of the flow in a fluid medium. Reynolds number is the ratio of the force of inertia to the viscous force under different velocities in a fluid medium. Reynolds flow is used to consider whether the flow is laminar or turbulent. Reynolds number is denoted by ‘Re’.

By definition Reynolds number is – 

\(R_e = \frac {inertia force}{viscous force} = \frac{pAV^2}{\mu VA/l} = \frac{pVl}{\mu}\)

Where we know, 

Inertia force Fi = mass * acceleration

=

\(Inertia force F_r = pl^2V^2\)

\(Viscous force(F_v) = Shear stress X area = \mu \frac{velocity}{length} X Area\)

= 

\(\mu VA/l\)

Where,

• Re is Reynolds number 
• P is the Density of the fluids 
• U is the Speed of the flow 

• L is the Linear Dimensional Characteristics 
• μ is the Dynamic Viscosity of the fluid 

So, as per the above elucidation, 

• Reynolds number is dependent on inertia and viscous force. On the other hand, the Froude number is dependent on inertia and gravity force. 
• Both being dimensionless forces are capable of comparison. Where Froude number is used to know the measure of comparison of waves which makes resistance between bodies, Reynolds number comes into the picture on considering whether the flow is laminar or turbulent.

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Things to Remember 

• Froude number is a dimensionless quantity that was named after great scientist William Froude. It is used to elucidate the effect of gravity on the motion of the fluid. Froude number is defined as the ratio of inertia to the gravitational force. It is applicable in a fluid mechanical study in different mediums. 
• Dimensionless quantity is known to have no physical dimensions. It is highly used in the comparison of different quantities. Dimensionless quantity is measured as the ratio of quantities without dimensions to the ratio of quantities with dimensions.
• Froude number is described, mathematically as the ratio of inertia to the gravity force. So, the Froude number in the formula is -\(Fr = \frac{V}{\sqrt{gD}}\)

Where -

Fr is Froude Number 

V is Velocity 

g is Gravity 

D is Hydraulic Depth or depth of flow (I) 

Froude number can also be elucidated as -\(F_r = (\frac{inertia force}{gravity force})^{1/2}\)

Where we know inertia is -\(Inertia force F_r = p l^2V^2\)

And, Gravity force = mass * gravitational acceleration

 \(= pl^3g\)