What is the Formula of Froude Number?
The Froude number is a dimensionless physical parameter that explains the different flow management of an open channel flow. The Froude number formula basically gives a ratio of inertial and gravitational forces. In special cases of fluid dynamics where the gravitational force is of importance then the Froude number governs the entire required dynamic similarity. The Froude number formula was named after the physicist William Froude. The Froude number relies on the speed-length ratio and it is the most important part of fluid dynamics. The Froude number is found to be identical to the Mach number.
At the same time, while dealing with the theoretical fluid dynamics, it is often noticed that the Froude number is not considered often. The reason behind this is, regularly the expressions that are considered in the high Froude limit in the case of the negligible or weak external field. In this article, we will focus on the Froude number formula, Froude number, and Froude definition with the help of solved problems.
Froude Number
The Froude Number is important in fluid dynamic calculations where the weight (we know that weight of the body is equal to the gravitational force) of the fluid under consideration is one of the important forces. Usually, this is the condition for free surfaces such as cold windows and hot radiators or flow in open conduits such as any regular water channels, sewer pipes, etc... Froude no. is generally used while calculating the momentum transfer in general and open channel flow and wave and surface behaviour in particular.
The Froude Number is of importance while analyzing flow in spillways, weirs, channel flows, rivers, and in ship design. If you study naval architecture, the Froude number is said to be a significant figure which is used in the estimation and determination of the resistance of the partially submerged object or any body which is moving through the water. If you observe in an open channel flows, the ratio of the flow velocity to the square root of the product of acceleration due to gravity to the depth of the flow.
William Froude utilized a series of scale models to determine the resistance of each module when it is towed at the given speed or the velocity. The concept of Froude no was first introduced by Frederic Reech and is applied for testing ships and propellers for understanding the dynamics. But the speed-length ratio was first introduced and derived by Fraude in his Law of Comparison. Where he showed that the dimensional terms consist of the speed-length ratio is equal to \[\frac{u}{\sqrt{\text{level of the water line}}}\].
Now, let us have a look at the Froude number formula. The Froude number formula is given by the following expression:
\[\Rightarrow F_r = \frac{v}{\sqrt{gl}} \]
Where,
\[v\] - The velocity of the fluid flow
\[g\] - The acceleration due to gravity
\[l\] - Depth of flow
There, the Froude number or the Froude no. is defined as the ratio of inertial force to the weight of the fluid element.
Examples:
A fluid element is flowing through a tube with a mean depth of 0.2m with a velocity of 6m/s. Then, determine the Froude number associated with the given fluid system?
Sol:
Given,
The mean depth of the given fluid flow \[= l =0.2m\]
The velocity of the flow of fluid \[= v = 6m/s\]
The acceleration due to gravity \[ = g = 9.8 m/s^2\]
Now, we are asked to determine the Froude number associated with the flow of a given fluid element. We know that the Froude number can be calculated using the following Froude number formula. We have:
\[\Rightarrow F_r = \frac{v}{\sqrt{gl}} \].....(1)
Where,
\[v\] - The velocity of the fluid flow
\[g\] - The acceleration due to gravity
\[l\] - Depth of flow
Substituting all the known values in the Froude no. equation and simplify. We get:
\[\Rightarrow F_r = \frac{v}{\sqrt{gl}} \]
\[\Rightarrow F_r = \frac{6}{\sqrt{9.8+0.2}} = \frac{6}{1.4} \]
\[\Rightarrow F_r = 4.285 \approx 4.3 \]
Therefore, the Froude number associated with the flow of a given fluid element is 4.3
The Froude number associated with the flow of a fluid element is 5 and the fluid is flowing through a tube with a mean depth of 0.4 m. Then, determine the velocity of the flow of the given fluid?
Sol:
Given,
The mean depth of the given fluid flow \[= l = 0.4m\]
The Froude number associated with the fluid flow \[= F_{r} = 5 \]
The acceleration due to gravity \[= g = 9.8 m/s^2\]
Now, we are asked to determine the velocity of the flow of a given fluid element. We know that the Froude number can be calculated using the following Froude number formula. We have:
\[\Rightarrow F_r = \frac{v}{\sqrt{gl}} \].....(1)
Where,
\[v\] - The velocity of the fluid flow
\[g\] - The acceleration due to gravity
\[l\] - Depth of flow
Rearranging equation (1) for the velocity and Substituting all the known values in the above equation and simplify. We get:
\[\Rightarrow F_r = \frac{v}{\sqrt{gl}} \]
\[\Rightarrow v = F_r \times \sqrt{gl} \]
\[\Rightarrow v = (5)\times \sqrt{(9.8+0.4)} = 5\times 1.98 \]
\[\Rightarrow v = 9.9 m/s \]
Therefore, the velocity of the flow of a given fluid element is \[9.9 m/s \].
Estimate the velocity of the fish moving through the water of the mean depth 0.5m and the associated Froude number for the flow of the fish is 1.2.
Sol:
The mean depth of the given fluid flow \[=l=0.5m\]
The Froude number associated with the fluid flow \[= F_r =1.2 \]
The acceleration due to gravity \[= g = 9.8 m/s^2\]
Now, we are asked to determine the velocity of the flow of a given fluid element. We know that the Froude number can be calculated using the following Froude number formula. We have:
\[\Rightarrow F_r = \frac{v}{\sqrt{gl}} \].....(1)
Where,
\[v\] - The velocity of the fluid flow
\[g\] - The acceleration due to gravity
\[l\] - Depth of flow
Rearranging equation (1) for the velocity and Substituting all the known values in the above equation and simplify. We get:
\[\Rightarrow F_r = \frac{v}{\sqrt{gl}} \]
\[\Rightarrow v = (1.2) \times \sqrt{(9.8+0.5)} = 1.2 \times 2.214 \]
\[\Rightarrow v = 2.56 m/s \]
Therefore, the velocity of the fish flowing through the water is \[2.56 m/s \].
FAQs on Froude Number Formula
1. What is the significance of the Froude number?
The Froude number is basically a measurement of bulk flow characteristics of fluid flow such as waves, sand bedforms, flow/depth interactions at a cross-section or between boulders. The denominator of the Froude number formula represents the speed of a small wave on the water surface relative to the speed of the water, called wave celerity.
2. Why is the Froude number is important?
The Froude number plays a vital role in the fluid dynamics where the gravitational force of the important. The major significance of Froude Number is in mainly analyzing the flow in spillways, channel flows, rivers, weirs, and even in the ship design.
