Froude Number

In the dimension analysis, the dimensionless quantity is the quantity that does not have physical dimensions. These dimensionless quantities are defined as the product or ratio of the quantities with dimensions. For example, strain is the measure of deformation, in other words, it can be written as the ratio of change in length over the initial length. Since the unit is ‘L’ in both cases thus a strain is dimensionless. 

The dimensionless numbers are used to analyze the prototype models. These numbers are Reynolds numbers, Weber’s number, and Froude’s number. In situations where the gravitational force is important, the Froude number governs dynamic similarity. Some of the examples include, flow through the open channels, flow over the spillway of the dam, etc. 

Froude Number Formula

Froude number is represented by Fr, in fluid mechanics, and in hydrology, Fr is a dimensionless quantity that is used in the indication of the influence of gravity on the motion of fluids. It is expressed as,

Froude No = \[\frac{v}{\sqrt{gd}}\]

Here, ‘d’ represents the depth of flow, 

‘g’ is the acceleration due to gravity,

‘v’ is the celerity or gravity of the small surface wave. 

When Froude No. is at a critical point that is Fr = 1, then the velocity of the flow is equal to the velocity of surface waves, which is also known as the Critical flow Froude number. 

If the Fr is less than one, then this means that the small surface waves can move in an upstream direction or it is considered as fluvial motion. 

If Fr is greater than one, then this means that the small surface waves can move in the downstream direction or it is considered as torrential flow motion. The Froude number enters into the hydraulic jump that occurs in certain conditions and along with the Reynolds number. 

Froude Definition

Froude’s number is named after the scientist William Froude. It depends on the speed-length ratio. It has some analogy with the Mach number. In the case of theoretical fluid dynamics, it is not considered frequently because usually the equations are considered in the high Froude limit in the case of the negligible external field. 

In naval architecture, the Froude number is the significant figure that is used in the determination of the resistance of the partially submerged object that is moving through the water. In the 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 used a series of scale models to measure the resistance of each module when it is towed at the given speed. This concept was first introduced by Frederic Reech and is used for testing ships and propellers. But the speed-length ratio was originally derived by Fraude in his Law of Comparison. Where the dimensional terms consist of 

Speed - length ratio = \[\frac{u}{\sqrt{LWL}}\]

Where u is the flow speed and LWL indicates the length of the waterline. It is converted into a non-dimensional term. It is also known as the Reech–Froude number in the region of France.

Froude Number Applications

1. Cauchy Momentum Equation: The characteristic length r0 and characteristic velocity u0 are to be defined to make the equations dimensionless. The dimensionless variables should be of one order. 

ρ* ≡ \[\frac{\rho }{\rho _0}\], u* ≡ \[\frac{u}{u_0}\], r* ≡ \[\frac{r}{r_0}\], t* ≡ \[\frac{u_0}{r_0}\]t, g* ≡ \[\frac{g}{g_0}\], σ* ≡ \[\frac{\sigma }{\sigma _0}\] 

If we substitute these inverse relations in the Euler momentum equations we get,

Froude No = \[\frac{u_0}{\sqrt{g_0r_0}}\]

Thus Cauchy momentum equation is,

\[\frac{Du}{Dt}\] + Eu\[\frac{1}{\rho }\] ▽ . σ = \[\frac{1}{Fr^2}\]g 

In the high Froude limit, the Cauchy type of equations is known as free equations, where Fr → ∞ and in the lower Euler limit, where Eu → 0, the Cauchy momentum equation becomes in the homogeneous equation called the Burgers equation.

\[\frac{\partial u}{\partial t}\] + ▽ . (\[\frac{1}{2}\]u ⊗ u) = \[\frac{1}{Fr^2}\]g

2. Euler Momentum Equation: It is a Cauchy momentum equation along with the Pascal law with the relation of stress constitutive.

3. Incompressible Navier–Stokes Momentum Equation: It is a Cauchy momentum equation along with the Pascal law and Stoke’s law with the relation of stress constitutive.

4. Ship Hydrodynamics: In the applications of marine hydrodynamics, Froude number is usually referred to as Fn, 

FnL = \[\frac{u}{\sqrt{gL}}\]

Here, u is considered as the relative flow velocity in between the sea and ship, g is the acceleration due to gravity and L is the length of the ship at the level of the waterline, it is also denoted as Lwl. It is the important parameter that is calculated for the resistance or ship drag in respect to the wave-making resistance.

In the planning crafts, while the waterline length is too speed-dependent, the Froude number can be defined as displacement Froude number, and thus the reference length is taken from the volumetric displacement of the hull, it is represented as:

FnV = \[\frac{u}{\sqrt{g^3}\sqrt{v}}\]

5. Shallow Water Waves: The shallow waterways such as hydraulic jumps and tsunamis, U is considered as the average flow velocity, where this U is averaged to the cross-section that is perpendicular to the flow direction. Thus the wave velocity c is defined as:

c = \[\sqrt{g\frac{A}{B}}\], hence the Froude number in the shallow water will be, 

Fr = \[\frac{u}{\sqrt{g\frac{A}{B}}}\]

For the uniform depth and rectangular cross-section, the Froude number is given as,

Fr = \[\frac{u}{\sqrt{gd}}\]

If Fr ≈ 1 then the flow is critical, 

If the Fr < 1 then the flow is Subcritical, 

If the Fr > 1 then the flow is Supercritical. 

6. Wind Engineering: In dynamically sensitive structures, considering the wind effects is necessary for the simulation of the combined effect of the vibrating mass along with the fluctuating force of the wind. When simulating the hot smoke plumes when combined with the natural wind, the scaling of the Froude number is necessary for the maintenance of the correct balance between the buoyancy forces and the momentum of the wind. 

Hydraulic Jump

In the science of hydraulics, the hydraulic jump is the phenomenon that is frequently observed in the case of open channel flows that include rivers and spillways. When the liquid is at a high velocity and if it discharges to the zone of lower velocity then the rise occurs in the region of the liquid surface. This rapidly flowing water abruptly slows down and increases its height, thus converting some of the flow from kinetic energy to potential energy during this conversion some of the energy is lost in the form of heat of turbulence. 

This phenomenon is dependent on the initial speed of the fluid, if the initial speed is said to be lower than that of the critical speed then there is no jump observed in the liquid, thus in such cases, the transition appears as an undulating wave. The hydraulic jump is seen in both dynamic form and stationary form. The stationary form is known as a hydraulic jump, and in the dynamic form, it is known as positive surge or hydraulic jump of translation. 

The final equation in the representation of the hydraulic jump consists of Froude No, it is also known as Belanger equation, it is represented as,

\[\frac{h_1}{h_2}=\frac{\sqrt{1+8Fr^2}-1}{2}\]

If the Fr value is greater than one, then the initial flow represents the supercritical flow, and if the Fr is less than one, then it represents the subcritical flow. 

The dam should be designed in such a way that the energy of the fast-flowing stream of the spillway should be dissipated partially to avoid the erosion of the streambed downstream of the spillway at the end this leads to failure of the dam. An apron is designed to withstand the hydraulic forces and prevent the local cavitation to avoid erosion. Thus to design these the engineers select a point where the hydraulic jump will occur. To trigger the hydraulic jump, the apron is designed in such a way that the flat slope of this retards the flow of the water from the spillway. There are two methods to induce the jump:

If the downstream channel restricts the downstream flow, then the water backs up onto the foot of the spillway, thus the downstream water level is used to determine the location of the hydraulic jump.

At some distance, the spillway continues to drop the flow, but this flow will be changed to prevent the support of the supercritical flow since the depth of the lower subcritical flow region is required to identify the location of the jump. 

In both of these cases, the final depth of the water can be determined by downstream characteristics, 

\[h_0=\frac{h_1}{2}(-1+\sqrt{1+8Fr_2^2}\]

Where Fr represents the upstream Froude number, g is the acceleration due to gravity, h is the height of the fluid.

Based on the Froude number hydraulic jump is classified as:

• If the Fr is in between 1.0 to 1.7, then it is known as undular jump.

• If the Fr is between 1.7 to 2.5, then it is known as a weak jump.

• If the Fr is in between 2.5 to 4.5 it is known as an oscillating jump.

• If the Fr is in between 4.5 to 9.0 then it is known as a steady jump.

Froude's model law :

A model's Froude number should be the same as a prototype's Froude number. It is used in situations where gravity is simply acting as an overriding force to govern flow as well as inertia force.

Uses of Froude Number :

Following are some of the uses of Froude Number:

• It is significant in fluid dynamics problems when the fluid's weight is a substantial force.

• It is taken into account while examining flow in spillways, channel flows, weirs, rivers, and ship design.

• It is useful for calculating hydraulic jumps and building hydraulic structures when gravity and inertial forces are in control.

• Bedforms' characteristics shall be determined

• It is used to overcome the resistance of an object moving through water.

• Fr. compares the resistance caused by a wave among bodies of various shapes and sizes.

• It is useful in low-water waves, wind engineering, and ship hydrodynamics.

Conclusion

In between the bodies of various sizes and shapes, the Froude number is used to determine wave-making resistance. In the case of free surface flow, the nature of the flow is determined by the value of the Froude number that is greater than or less than the unity. The line of critical flow is usually in the kitchen or bathroom sinks. 

When we keep it unplugged and allow the faucet to run, the place where the stream hits the sink, that flow is considered as the supercritical flow. The stream hugs the surface and moves very quickly. The outer edge of the flow pattern is subcritical, the boundary between these two areas is known as the hydraulic jump. This jump starts only when the Froude number is 1.