Reynolds Number Calculator

Calculate the Reynolds number to determine if a fluid flow is laminar or turbulent.

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Enter the fluid density in kg/m³

Enter the fluid velocity in m/s

Enter the diameter in meters

Enter the dynamic viscosity in Pa·s

Table of Contents

1 Understanding Reynolds Number
2 Reynolds Number Formula
3 How to Calculate
4 Flow Regimes
5 Practical Examples
Fundamentals

Understanding Reynolds Number

The Reynolds number is a fundamental dimensionless parameter in fluid mechanics that quantifies the relationship between inertial forces and viscous forces in a flowing fluid. Named after Osborne Reynolds, who popularized its use in the 1880s, this number serves as a critical indicator to predict flow patterns and behavior.

Historical Background

The concept was first introduced by George Stokes in 1851, but it was Osborne Reynolds who conducted groundbreaking experiments in 1883 that demonstrated its practical significance. Reynolds used a simple yet ingenious apparatus with dyed water flowing through a glass tube to visualize the transition from laminar to turbulent flow. The dimensionless number was later named after Reynolds by Arnold Sommerfeld in 1908.

Physical Significance

The Reynolds number represents the ratio of inertial forces (which drive fluid motion) to viscous forces (which resist motion). This relationship determines whether a flow will be:

  • Laminar: At low Reynolds numbers, viscous forces dominate, resulting in smooth, orderly flow with fluid layers sliding past each other in parallel paths.
  • Transitional: At intermediate Reynolds numbers, the flow begins to show instabilities and oscillations between laminar and turbulent characteristics.
  • Turbulent: At high Reynolds numbers, inertial forces dominate, creating chaotic, irregular flow patterns with eddies, vortices, and other flow instabilities.

Critical Values and Applications

The critical Reynolds number that marks the transition from laminar to turbulent flow varies depending on the flow geometry:

  • For flow in a pipe: transition typically begins at Re ≈ 2300
  • For flow over a flat plate: transition occurs around Re ≈ 500,000
  • For flow around a sphere: transition occurs around Re ≈ 2×105

The Reynolds number finds applications across diverse fields of science and engineering:

  • Civil engineering: designing water distribution systems and sewage networks
  • Aerospace engineering: analyzing airflow around aircraft wings and bodies
  • Chemical engineering: designing reactors and mixing systems
  • Biomedical engineering: studying blood flow in vessels and artificial organs
  • Environmental engineering: modeling atmospheric and oceanic flows
Did You Know?
  • The Reynolds number in the human aorta during normal blood flow is approximately 1,000, indicating mostly laminar flow.
  • For commercial aircraft, the Reynolds number may exceed 107, strongly turbulent flow.
  • The Reynolds number for swimming bacteria is around 10-4, indicating a completely different fluid dynamics regime where viscous forces dominate.
Concept

Reynolds Number Formula

The Reynolds number is a dimensionless quantity used to predict flow patterns in different fluid flow situations.

Formula:
Re = (ρ × v × L) / μ

Where:

  • Re = Reynolds number (dimensionless)
  • ρ = Fluid density (kg/m³)
  • v = Flow velocity (m/s)
  • L = Characteristic length (m)
  • μ = Dynamic viscosity (Pa·s)
Steps

How to Calculate

To calculate the Reynolds number, follow these steps:

  1. 1
    Determine the fluid density (ρ)
  2. 2
    Measure the flow velocity (v)
  3. 3
    Determine the characteristic length (L)
  4. 4
    Find the dynamic viscosity (μ)
  5. 5
    Multiply density, velocity, and length, then divide by viscosity
Advanced

Flow Regimes

The Reynolds number helps determine the type of flow:

  • Re < 2300: Laminar flow (smooth, orderly flow)
  • 2300 ≤ Re < 4000: Transitional flow
  • Re ≥ 4000: Turbulent flow (chaotic, irregular flow)
Important Notes:
  • These values are approximate and may vary depending on the specific application
  • The transition between flow regimes is not abrupt but gradual
  • Other factors like surface roughness can affect the transition
Examples

Practical Examples

Example 1 Water in a Pipe

Calculate the Reynolds number for water flowing through a 2 cm diameter pipe at 1 m/s.

ρ = 1000 kg/m³

v = 1 m/s

L = 0.02 m

μ = 0.001 Pa·s

Re = (1000 × 1 × 0.02) / 0.001 = 20,000

Example 2 Air Flow

Calculate the Reynolds number for air flowing over a 1 m wide plate at 5 m/s.

ρ = 1.225 kg/m³

v = 5 m/s

L = 1 m

μ = 1.81 × 10⁻⁵ Pa·s

Re = (1.225 × 5 × 1) / (1.81 × 10⁻⁵) = 338,398

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