Froude Number Calculator

What is Froude Number?

The Froude number (Fr) is a dimensionless number which is used to characterizes the flow of fluids. It is named after the British engineer William Froude.

The Froude number is defined as the ratio of inertia force to the gravitational forces in fluid dynamics. This number is important in the study of open channel flows, ship hydrodynamics, and the behavior of flow around objects.

Mathematically, it is defined as:

\[\text{Fr} =\frac{\text{Inertial Force}}{\text{Gravitational Force}}\]

\[\text{Fr} = \frac{ \rho u^2 L^2}{\rho g L^3} = \frac{U^2}{g L}\]

\[\text{Fr} = \frac{U}{\sqrt{gL}}\]

where:

\(U\) is the characteristic velocity of the flow
\(g\) is the acceleration due to gravity
\(L\) is the characteristic length

This Calculator is a web tool designed to compute the Froude number. User are able to input values for velocity (U), and characteristic length (L), and then it performs the calculation which helps in understanding flow regime in open channel and around objects.

Froude Number Calculator

Related: Other Dimensionless Numbers

Significance of Froude Number

The Fr number is used to categorize and understand different flow regimes based on their velocity relative to the speed of surface waves. It is helpful for engineers and scientists to predict and design structures and systems that interact with flowing fluids, ensuring safety, efficiency, and optimal performance.

Case 1. Subcritical Flow or Tranquil Flow (Fr < 1)

In this regime, the flow velocity is slower than the speed of gravity waves that would propagate along the surface of the fluid. For example, A calm and slow-moving sections of the river where the flow is controlled by downstream conditions.

Case 2. Critical Flow (Fr = 1)

This is the point when the flow velocity is equal to the speed of the surface gravity waves. For example, Critical flow often occurs over weirs or sharp changes in the river bed where significant energy transformations occur.

Case 3. Supercritical Flow (Fr > 1)

The flow velocity is faster than the speed at which surface gravity waves propagate. For example, Supercritical flow can occur in rapidly flowing rivers or when fluid accelerates rapidly downhill.

Froude Number Applications

The Froude Number (Fr) is an important dimensionless parameter in fluid mechanics, which is widely used to characterize and analyze different flow regimes across various engineering and environmental applications.

In Ship Design and Naval Architecture, Fr number helps to determine the optimal speed of a ship and ensures stable operation in different sea conditions. A low Fr Number results in lower wave resistance (subcritical flow), whereas high Fr number leads to higher wave resistance (supercritical flow).
The stability of hydraulic structures, such as dams and spillways, Supercritical flows (Fr > 1) can cause turbulent conditions that might damage these structures, requiring careful design considerations.
In Atmospheric and Oceanic flows, the Fr Number is used to analyze the stability of stratified flows, such as the flow of air over mountains or the flow of ocean currents over underwater topography.

Python Code for Froude Number for Open Channel

This Python script helps user to visualize how the Froude number (Fr) varies with hydraulic depth in an open channel flow at different velocities. Each velocity is represented by a line plot where colors distinguish different velocities, and annotations on the plot indicate the flow regime at specific depths.

Note: This Python code solves the specified problem. Users can copy the code and run it in a suitable Python environment. By adjusting the input parameters, and observe how the output changes accordingly.

import numpy as np
import matplotlib.pyplot as plt

def calculate_froude_number(V, g, depth):
    return V / np.sqrt(g * depth)

# Define parameters
g = 9.81  # Acceleration due to gravity (m/s^2)
depths = np.linspace(0.1, 5, 100)  # Depths in the open channel (m)
velocities = [1, 2, 3, 4, 5]  # Velocities (m/s)

# Plot Froude number vs depth for each velocity with different colors
plt.figure(figsize=(12, 6))

for i, V in enumerate(velocities):
    froude_numbers = calculate_froude_number(V, g, depths)
    flow_types = []

    for fn in froude_numbers:
        if fn < 1:
            flow_types.append('Subcritical Flow')
        elif fn == 1:
            flow_types.append('Critical Flow')
        else:
            flow_types.append('Supercritical Flow')

    color = plt.cm.jet(i / len(velocities))  # Jet colormap for distinct colors
    plt.plot(depths, froude_numbers, label=f'V = {V} m/s', color=color)

    # Annotate flow regime at specific depths
    annotate_depths = [0.5, 1.5, 3.0]  # Depths to annotate
    for depth in annotate_depths:
        index = np.argmin(np.abs(depths - depth))  # Find index closest to desired depth
        plt.annotate(flow_types[index], (depth, froude_numbers[index]),
                     textcoords="offset points", xytext=(5,10), ha='center', color=color)

plt.xlabel('Hydraulic Depth (m)')
plt.ylabel('Froude Number')
plt.title('Froude Number vs Depth in Open Channel Flow')
plt.axhline(y=1, color='black', linestyle='--', label='Critical Flow (Froude Number = 1)')
plt.legend()
plt.grid(True)
plt.show()

Output:

froude number vs hydraulic depth in open channel flow

Resources

“Fluid Mechanics” by Frank M. White
“Introduction to Fluid Mechanics” by Robert W. Fox, Alan T. McDonald, and Philip J. Pritchard
“Principles of Heat and Mass Transfer” by Frank P. Incropera and David P. DeWitt
Python.org – The official Python website offers tutorials, documentation, and resources for learning Python.

Disclaimer: The Solver provided here is for educational purposes. While efforts ensure accuracy, results may not always reflect real-world scenarios. Verify results with other sources and consult professionals for critical applications. Contact us for any suggestions or corrections.

Froude Number Calculator – Significance and Applications

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What is Froude Number?