Reynolds Number Calculator for a Circular Pipe

What is Reynolds Number?

The Reynolds number is a dimensionless quantity used in fluid dynamics to predict the type of flow pattern, whether it’s laminar or turbulent, in a pipe. It’s defined by the ratio of inertial forces to viscous forces.

The formula for Reynolds number is given by:

\[ Re = \frac{\rho V D}{\mu} \]

where:

• \( Re \) is the Reynolds number
• \( \rho \) is the density of the fluid
• \( V \) is the velocity of flow
• \( D \) is the pipe diameter
• \( \mu \) is the viscosity of the fluid

For the circular pipe, if Re calculated is high (greater than 2000), then the flow through the pipe is said to be turbulent. If Re is low (less than 2000), the flow is said to be laminar.

Related: Newton’s Law of Viscosity Calculator – Dynamic Viscosity

Related: Hagen-Poiseuille Equation Calculator / Poiseuille’s Law Solver

Reynolds Number Calculator

This calculator computes the Reynolds number for a fluid flow based on user inputs for fluid density (ρ), flow speed (u), characteristic length (L), and dynamic viscosity (μ). After clicking the “Calculate” button, user gets calculated number and a message indicating the type of flow (laminar, turbulent, or transitional).

Note: User can choose the characteristic length for the given geometry of the flow situation.

Related: Other Fluid Mechanics Calculators

Characteristic Length Table for Reynolds number calculation

Flow Situation	Characteristic Length (L)	Description
Flow over a flat plate	Length of the plate	The characteristic length is the length of the plate over which the fluid is flowing.
Flow inside a pipe or duct	Hydraulic diameter	The hydraulic diameter is defined as four times the cross-sectional area of the pipe divided by the wetted perimeter of the pipe. It is used as the characteristic length for flow inside a pipe or duct.
Flow around a cylinder	Diameter of the cylinder	The characteristic length is the diameter of the cylinder around which the fluid is flowing.
Flow around a sphere	Diameter of the sphere	The characteristic length is the diameter of the sphere around which the fluid is flowing.

Reynolds Number Derivation

The Reynolds number is defined as the ratio of inertial forces to viscous forces within a fluid.

\(R_e = \frac{inertial force}{viscous force}\)

Let’s consider a fluid of density (\( \rho \)) is flowing with velocity (\( v \)) over a characteristic length (\( L\)).

The mass of the fluid per second, ( \( \Delta m \)), is given by the volume of fluid flowing per second multiplied by the density, which is (\( A \cdot v \cdot \rho \)).

The inertial force per unit area is then given by the rate of change of momentum divided by the area, which is

\( \frac{\Delta m \cdot v}{A} = \frac{A \cdot v \cdot \rho \cdot v}{A} = \rho \cdot v^2 \).

The viscous force, ( F ), is given by (\( \eta \cdot A \cdot \frac{v}{L} \)), where ( \(\frac{v}{L}\) ) is the velocity gradient between the layers of the liquid flow.

The viscous force per unit area is then

\( \frac{F}{A} = \eta \cdot \frac{v}{L}\)

Therefore, substituting the above expressions, we get:

\( R_e = \frac{\rho \cdot v^2}{\eta \cdot \frac{v}{L}} = \frac{v \cdot \rho \cdot L}{\eta} \)

This is the mathematical expression, that provides insight into the flow regime of a fluid, whether it’s laminar or turbulent. The characteristic length ( L ) is chosen based on the specific flow situation as shown in table above.

Example Problem on Reynolds Number

Determine the flow of fluid having a relative density of \(100 \, \text{kg/m}^3\), the viscosity of \(0.5 \, \text{Ns/m}^2\) with a velocity of \(5 \, \text{m/s}\) through a pipe of \(0.2 \, \text{m}\).

Given:

• Velocity of fluid, \(V = 5 \, \text{m/s}\)
• Diameter of pipe, \(D = 0.2 \, \text{m}\)
• Relative density of fluid, \(p = 100 \, \text{kg/m}^3\)
• Viscosity of fluid, \(\mu = 0.5 \, \text{Ns/m}^2\)

using formula \( \text{Re} = \frac{pVD}{\mu} \)

substituting given values in the formula we get:

\( \text{Re} = \frac{(100 \, \text{kg/m}^3) \times (5 \, \text{m/s}) \times (0.2 \, \text{m})}{0.5 \, \text{Ns/m}^2} \)

\(\text{Re} = 200 \)

Since the Re is less than \(2000\), the flow of liquid is laminar.

Python Code for Reynolds Number Calculation

This Python code helps user to calculate and visualizes the Reynolds number to predict flow patterns. This code plots the Reynolds number against three parameters: velocity, characteristic length, and viscosity, and showing them as subplots. The regions of laminar and turbulent flow regimes are highlighted on each plot with the variation in colors corresponding to different Re number ranges.

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, users can observe how the output changes accordingly.

import numpy as np
import matplotlib.pyplot as plt

def reynolds_number(density, velocity, length, viscosity):
    """
    Calculate Reynolds_number.

    Parameters:
    density (float): Fluid density (kg/m^3).
    velocity (float): Flow velocity (m/s).
    length (float): Characteristic length (m).
    viscosity (float): Fluid viscosity (Pa*s).

    Returns:
    reynolds_number (float): Reynolds number.
    """
    return density * velocity * length / viscosity

def plot_reynolds_vs_velocity(density, length, viscosity):
    """
    Plot Reynolds number against velocity.

    Parameters:
    density (float): Fluid density (kg/m^3).
    length (float): Characteristic length (m).
    viscosity (float): Fluid viscosity (Pa*s).
    """
    velocities = np.linspace(0.1, 10, 100)
    reynolds_numbers = np.array([reynolds_number(density, velocity, length, viscosity) for velocity in velocities])

    plt.subplot(3, 1, 1)
    plt.plot(velocities, reynolds_numbers)
    plt.fill_between(velocities, 0, reynolds_numbers, where=(reynolds_numbers <= 2000), color='blue', alpha=0.3)
    plt.fill_between(velocities, 0, reynolds_numbers, where=(reynolds_numbers > 2000), color='red', alpha=0.3)
    plt.xlabel('Velocity (m/s)')
    plt.ylabel('Reynolds Number')
    plt.title('Reynolds Number vs Velocity')
    plt.grid(True)

def plot_reynolds_vs_length(density, velocity, viscosity):
    """
    Plot Reynolds_number against characteristic length.

    Parameters:
    density (float): Fluid density (kg/m^3).
    velocity (float): Flow velocity (m/s).
    viscosity (float): Fluid viscosity (Pa*s).
    """
    lengths = np.linspace(0.1, 5, 100)
    reynolds_numbers = np.array([reynolds_number(density, velocity, length, viscosity) for length in lengths])

    plt.subplot(3, 1, 2)
    plt.plot(lengths, reynolds_numbers)
    plt.fill_between(lengths, 0, reynolds_numbers, where=(reynolds_numbers <= 2000), color='blue', alpha=0.3)
    plt.fill_between(lengths, 0, reynolds_numbers, where=(reynolds_numbers > 2000), color='red', alpha=0.3)
    plt.xlabel('Characteristic Length (m)')
    plt.ylabel('Reynolds Number')
    plt.title('Reynolds Number vs Characteristic Length')
    plt.grid(True)

def plot_reynolds_vs_viscosity(density, velocity, length):
    """
    Plot Reynolds_number against viscosity.

    Parameters:
    density (float): Fluid density (kg/m^3).
    velocity (float): Flow velocity (m/s).
    length (float): Characteristic length (m).
    """
    viscosities = np.linspace(0.1, 1, 100)
    reynolds_numbers = np.array([reynolds_number(density, velocity, length, viscosity) for viscosity in viscosities])

    plt.subplot(3, 1, 3)
    plt.plot(viscosities, reynolds_numbers)
    plt.fill_between(viscosities, 0, reynolds_numbers, where=(reynolds_numbers <= 2000), color='blue', alpha=0.3)
    plt.fill_between(viscosities, 0, reynolds_numbers, where=(reynolds_numbers > 2000), color='red', alpha=0.3)
    plt.xlabel('Viscosity (Pa*s)')
    plt.ylabel('Reynolds Number')
    plt.title('Reynolds Number vs Viscosity')
    plt.grid(True)

# Example usage:
density = 1000  # kg/m^3
velocity = 1  # m/s
length = 1  # m

plt.figure(figsize=(5, 7))
plot_reynolds_vs_velocity(density, length, velocity)
plot_reynolds_vs_length(density, velocity, length)
plot_reynolds_vs_viscosity(density, velocity, length)

plt.tight_layout()
plt.show()

Output:

Resources

1. “Fluid Mechanics” by Frank M. White
2. “Introduction to Fluid Mechanics” by Robert W. Fox, Alan T. McDonald, and Philip J. Pritchard
3. “Principles of Heat and Mass Transfer” by Frank P. Incropera and David P. DeWitt
4. 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.