The Reynolds Number

Learning Objectives

• Define Pipe Flow: Explain the concept of pipe flow and distinguish it from open channel flow.
• Identify Key Parameters: Identify the key parameters that influence pipe flow, such as velocity, pressure, and flow rate.
• Differentiate Flow Types: Differentiate between laminar and turbulent flow.
• Explain Reynolds Number: Explain the Reynolds number and its significance in determining flow regimes.
• Analyze Flow Characteristics: Analyze the factors that influence flow characteristics, including pipe diameter, fluid velocity, viscosity, and roughness of the pipe's interior surface.

Before we dive further, we need to understand first what Reynold's number is and how it is derived. The formula in the video used "L" for length while we use "D" for diameter in the equation above. We use this interchangeably here. 

There are few things we need to establish first to understand the derivation: 

1. Newton's Law of Acceleration (F=ma)

2. Density (we get mass from density, thus m=ρV), where V here is volume.

3. Acceleration (a=v/t), where v is velocity

4. Volumetric flow rate (Q= flow velocity x area) or Q=vA

Reynolds number is equal to the ratio of inertial force over viscous force.

Check out the video below by Nico Enaya from BSED Science, where he explains the derivation process.

Now that you understand the derivation process, it's time to practice it yourself and master the technique and take the quiz below.

Please complete this QUIZ before moving on.

If you don't have the access code, click HERE to request.

The Reynolds number (Re) is a dimensionless quantity used in fluid mechanics to characterize the flow regime of a fluid. It plays a crucial role in understanding and predicting the behavior of fluid flow in various situations, such as in pipes, around objects, and in open channels.

The Reynolds number helps determine whether a fluid flow is laminar or turbulent:

• Laminar Flow: At low Reynolds numbers (typically Re < 2300), the flow is smooth and orderly, with fluid particles moving in parallel layers or streamlines. Viscous forces dominate, and the flow is relatively stable.
• Transitional Flow: In the range of Reynolds numbers between approximately 2300 and 4000, the flow can transition from laminar to turbulent. This regime is less predictable and can fluctuate between both types of flow.
• Turbulent Flow: At high Reynolds numbers (typically Re > 4000), the flow becomes chaotic and irregular, with mixing and eddies. Inertial forces dominate, and the flow is highly unstable and complex.

Applications

The Reynolds number is widely used in various engineering and scientific fields to analyze and design systems involving fluid flow:

• Pipe Flow: In determining the flow regime within pipes, which affects pressure drop and flow rate.
• Aerodynamics: In studying the flow around aircraft wings, car bodies, and other objects to optimize design for reduced drag and improved performance.
• Hydraulics: In understanding the behavior of water flow in open channels, rivers, and spillways.
• Heat Transfer: In analyzing convective heat transfer processes, where fluid flow impacts the rate of heat exchange.

The Reynolds number is a fundamental dimensionless quantity in fluid mechanics that provides insight into the nature of fluid flow. By comparing inertial and viscous forces, it helps engineers and scientists predict whether a flow will be laminar or turbulent, which is crucial for designing efficient fluid systems and understanding natural fluid phenomena.

Sample Problem

Oil with a dynamic viscosity of 0.1 Pa·s and a density of 900 kg/m³ is flowing through a pipe with a diameter of 0.05 m at a velocity of 2 m/s. Calculate the Reynolds number for the flow.

Solution

Oil with a dynamic viscosity of 0.1 Pa·s and a density of 900 kg/m³ is flowing through a pipe with a diameter of 0.05 m at a velocity of 2 m/s. Calculate the Reynolds number for the flow.

Solution:

Given:

• Dynamic viscosity of oil, $$\mu = 0.1 \, \text{Pa·s}$$
• Density of oil, $$\rho = 900 \, \text{kg/m}^3$$
• Diameter of the pipe, $$D = 0.05 \, \text{m}$$
• Velocity of the oil, $$v = 2 \, \text{m/s}$$

The Reynolds number (\(Re\)) is calculated using the formula:

$$Re = \frac{\rho \cdot v \cdot D}{\mu}$$

Substituting the given values:

$$Re = \frac{900 \, \text{kg/m}^3 \cdot 2 \, \text{m/s} \cdot 0.05 \, \text{m}}{0.1 \, \text{Pa·s}}$$

Simplifying the calculation:

$$Re = \frac{900 \cdot 2 \cdot 0.05}{0.1} = \frac{90}{0.1} = 900$$

So, the Reynolds number for the flow is 900.

Flow Regime Problems

1. A fluid with a viscosity of \(0.001 \, \text{Pa} \cdot \text{s}\) flows through a pipe with a diameter of \(0.05 \, \text{m}\). The fluid velocity is \(0.2 \, \text{m/s}\). Calculate the Reynolds number and determine the flow regime. Use 1000 kg/m3 density.
2. Water at \(20^\circ \text{C}\) (with a viscosity of \(0.001 \, \text{Pa} \cdot \text{s}\)) flows through a pipe with a diameter of \(0.1 \, \text{m}\). The flow rate is \(0.01 \, \text{m}^3/\text{s}\). Calculate the Reynolds number and identify if the flow is laminar, transitional, or turbulent. Use 1000 kg/m3 density.
3. Oil with a viscosity of \(0.05 \, \text{Pa} \cdot \text{s}\) flows through a pipe with a diameter of \(0.02 \, \text{m}\). The fluid velocity is \(1.5 \, \text{m/s}\). Determine the Reynolds number and classify the flow regime. Use 900 kg/m3 density.
4. Air at \(25^\circ \text{C}\) (with a viscosity of \(1.8 \times 10^{-5} \, \text{Pa} \cdot \text{s}\)) flows through a duct with a diameter of \(0.3 \, \text{m}\). The air velocity is \(10 \, \text{m/s}\). Calculate the Reynolds number and determine whether the flow is laminar, transitional, or turbulent. Use 1.1839 kg/m3 density.
5. A fluid with a viscosity of \(0.002 \, \text{Pa} \cdot \text{s}\) flows through a pipe with a diameter of \(0.08 \, \text{m}\). The fluid velocity is \(0.5 \, \text{m/s}\). Calculate the Reynolds number and identify the flow regime. Use 1000 kg/m3 density.