- Froude number calculation
- Froude number for an open pipe
- Flow types according to the Froude number
- Froude number and Reynolds number
- Worked example
- Solution
- References
The Froude number in hydraulics indicates the relationship between inertial forces and gravitational forces for a fluid. Therefore, it is a way of designating the following quotient:
Where N F is the notation for the Froude number, a dimensionless quantity given this name to honor the notable British naval architect and hydraulic engineer William Froude (1810-1879). Froude and his son experimented with dragging flat sheets through the water to estimate how resistant boats are to waves.
Figure 1. The Froude number is necessary to characterize the flow of water through an open channel, such as a ditch. Source: Pixabay.
In the action of the waves caused by a ship when sailing or the current on the pillar of a bridge, the forces of inertia and gravity are present.
The Froude number is particularly important in characterizing fluid flow in an open channel. An open pipe or channel is a conduit whose upper surface is open to the atmosphere. Examples abound in nature, in the form of rivers and streams.
And in man-made constructions we have:
-The gutters and drains in streets and buildings to conduct rainwater.
-Acequias for irrigation.
-Dumps and drains.
-Cooling channels for industrial machinery.
These are all examples of pipes open to the atmosphere, in which the Froude number must always be taken into account when characterizing the flow.
Froude number calculation
The quotient indicated at the beginning, between the forces of inertia and those of gravity, takes the following form, depending on the parameters of the fluid:
The previous equation or its square root is the Froude number:
Froude number for an open pipe
As explained at the beginning, the flow of water through channels open to the atmosphere is very frequent. For these cases, the calculation of the Froude number is carried out by applying the following formula:
Where y h is the hydraulic depth, v is the average flow velocity and g is the value of the acceleration of gravity. In turn, the hydraulic depth is calculated as follows:
In this formula, A represents the net cross-sectional area and T is the width of the free surface of the fluid, the one that is exposed to the atmosphere, at the top of the channel or pipe. It is valid for a rectangular channel or one that is wide enough and with constant depth.
It is important to note the fact that since NF is dimensionless, then the product g and h must be the square of a velocity. Indeed, it can be shown that:
With c o as the speed of propagation of a surface wave, analogous to the speed of sound in a fluid. Therefore the Froude number is also analogous to the Mach number, widely used to compare the speed of airplanes with that of sound.
Flow types according to the Froude number
Fluid flow in an open channel is classified into three regimes, according to the value of N F:
-When N F <1, there is a slow or subcritical movement.
-If N F = 1 the flow is called critical flow.
-Finally, if you have N F > 1, the movement is carried out in fast or supercritical regime.
Froude number and Reynolds number
The Reynolds number N R is another very important dimensionless quantity in fluid flow analysis, by which it is known when the fluid has laminar behavior and when it is turbulent. These concepts are applicable both to flows in closed pipes and in open channels.
A flow is laminar when the fluid moves smoothly and orderly in layers that do not mix. On the other hand, the turbulent flow is characterized by being chaotic and disorderly.
One way to find out if a water flow is laminar or turbulent is by injecting a stream of ink. If the flow is laminar, the ink stream flows separately from the water stream, but if it is a turbulent flow the ink mixes and dissipates into the water quickly.
Figure 2. Laminar flow and turbulent flow. Source: Wikimedia Commons. Seralepova
In this sense, when combining the effects of the Froude number with those of the Reynolds number, we have:
-Laminate subcritical: N R <500 and N F <1
-Subcritical turbulent: N R > 2000 and N F <1
-Supercritical rolling: N R <500 and N F > 1
-Supercritical turbulent: N R > 2000 and N F > 1
When the flows occur in the transition regions, it is more difficult to characterize them, due to their instability.
Worked example
A river 4 m wide and 1 m deep has a flow of 3 m 3 / s. Determine if the flow is subcritical or supercritical.
Solution
Finding the value of N F requires knowing the speed of the river current. The statement gives us the flow rate, also known as the volume flow rate, which depends on the cross-sectional area and the velocity v of the flow. It is calculated like this:
Where Q is the flow rate, A is the cross-sectional area and v is the velocity. Assuming a rectangular cross-sectional area:
Then the velocity v is:
The hydraulic depth in the case of the rectangular section pipe coincides with the depth, therefore, substituting values in the equation for N F, with y h = 1 m and g = 9.8 m / s 2 we have:
Since N F is less than 1, the flow has a subcritical behavior, that is, slow.
References
- Cimbala, C. 2006. Fluid Mechanics, Fundamentals and Applications. Mc. Graw Hill.
- Franzini, J. 1999. Fluid Mechanics with Application is in Engineering. Mc. Graw Hill.
- Mott, R. 2006. Fluid Mechanics. 4th. Edition. Pearson Education.
- White, F. 2004. Fluid Mechanics. 5th Edition. Mc Graw Hill.
- Wikipedia. Froude number. Recovered from: es.wikipedia.org.