Reynolds Number: Equation for Reynolds Number (With PDF)

Reynolds Number is a very important quantity for studying fluid flow patterns. It is a dimensionless parameter and widely used in fluid mechanics. Reynolds Number of a flowing fluid is defined as the ratio of inertia force to the viscous force of that fluid and it quantifies the relative importance of these two types of forces for given flow conditions. The concept of Reynold’s number was introduced by George Stokes in 1851. However, the name “Reynolds Number” was given with the name of the British physicist Osborne Reynolds, who popularized its use in 1883. The Reynolds number depends on the relative internal movement due to different fluid velocities. For fluid flow analysis, Reynold’s number is considered to be a pre-requisite.

Importance of Reynolds Number

Reynold’s Number (Re) is a convenient parameter that helps in predicting if a fluid flow condition will be laminar or turbulent. We know that Reynolds Number (Re)=inertia force/viscous force.

When viscous force dominates over the inertia force, the flow is smooth and at low velocities; the Reynolds Number value is comparatively less and flow is known as laminar flow. On the other hand, when inertia force is dominant, the value of Reynolds number is comparatively higher and the fluid flows faster at higher velocities and the flow is called turbulent flow. At low Reynolds Number Values (Re<2100) the viscous force is sufficient enough to keep fluid particles in line making the flow laminar which is characterized by smooth and constant fluid motion. While at large Reynold Number values (Re>4000), the flow tends to produce chaotic eddies, vortices, and other flow instabilities making the flow turbulent. With an increase in Reynolds Number the turbulence tendency of the flow increases.

Reynolds Number vs flow regimes
Fig. 1: Reynolds Number vs flow regimes

“2100<Reynolds Number (Re)<4000” indicates flow transition from laminar to turbulent and the flow consists of a mixed behavior. However, note that the value of Reynolds number (Re) at which turbulent flow begins is dependent on the geometry of the fluid flow, which is different for pipe flow and external flow.

Equation for Reynolds Number

Mathematically, The Equation for Reynolds number is represented as ρuD/μ

where 

  • ρ  is the fluid density Kg/m3)
  • D  is a length scale that characterizes the scale of the flow motions of interest (m)
  • u  is the fluid velocity (m/s)
  • μ  is the fluid dynamic viscosity (Pa.s or N.s/m2 or kg/m.s)
  • the term μ/ρ is known as kinematic viscosity, ν (m2/s)

Hence the formula for Reynold’s number can be written as Re=ρuD/μ=uD/ν

As the Reynolds Number is the ratio of two forces, there is no unit of Reynolds Number. So, Reynold’s Number is dimensionless.

Factors Affecting Reynolds Number

The main factors that govern the value of Reynolds Number are:

  • The fluid flow geometry
  • Flow velocity; with an increase in flow velocity the Reynolds number increases.
  • Characteristic Dimension; with an increase in characteristic dimension the Reynolds number increases.
  • Fluid Density; with a decrease in fluid density the Reynolds number value decreases.
  • Viscosity; with an increase in viscosity the value of the Reynolds number decreases.
Factors Affecting Reynolds Number
Fig. 2: Factors Affecting Reynolds Number

So, in one sentence we can conclude that Reynolds Number is directly proportional to Flow Velocity, Characteristic Dimension, and Fluid Density while inversely proportional to fluid viscosity.

Applications of Reynold’s Number

As Reynolds number is used for predicting laminar and turbulent flow, it is widely used as a design parameter for hydraulic and aerodynamic equipment. For the design of piping systems, aircraft wings, pumping system, scaling of fluid dynamic problems, etc Reynolds number serves as an important design tool. Reynold’s number is used to calculate the value of the drag coefficient. In the calculation of pressure drop and frictional losses, the Reynolds number plays an important role. The following diagram (Fig. 3), known as Moody chart provides a co-relation between friction factor, Reynold’s Number, and Relative roughness and widely used in solving fluid flow problems.

Reynolds number in Moody Chart
Fig. 3: Reynolds number in Moody Chart

Reynolds Number Value

The following table provides some typical Reynold Number values

Sr NoItemTypical Reynolds Number
1Laminar Flow<2100
2Turbulent Flow>4000
3Person Swimming4 × 106
4Blue Whale4 × 108
5Smallest fish1
6Atmospheric tropical cyclone1 x 1012
7Bacterium1 × 10−4
8Blood flow in brain1 × 102
9Blood flow in aorta1 × 103
10Fastest fish1 × 108
Typical Values of Reynolds Number (Reference: wikipedia.org)
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Anup Kumar Dey

I am a Mechanical Engineer turned into a Piping Engineer. Currently, I work in a reputed MNC as a Senior Piping Stress Engineer. I am very much passionate about blogging and always tried to do unique things. This website is my first venture into the world of blogging with the aim of connecting with other piping engineers around the world.

2 thoughts on “Reynolds Number: Equation for Reynolds Number (With PDF)

  1. I understood you explanation but could explain why the boundary layer thickness on a flat plate is proportional to the square root of the distance from the plate tip, and that the relative distance is inversely proportional to the Reynolds number ?

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