## Reynolds Number Formula

Reynolds number is used to determine the flow of fluid over or inside any object. Learn more about Reynolds number and its formula in this article.

Reynolds number is a very useful tool that is used mainly to predict the nature of the flow. By nature, it means whether the flow is turbulent or Laminar. But for that, we must also know what a turbulent or laminar kind of flow is.

A laminar flow is a flow when the fluid particles have a definite path and the path does not change. For example, when we open the tap, the water that flows in the first few inches of the pipe has a laminar flow.

A turbulent flow is a kind of flow where the fluid molecules do not have a definite path they randomly move in the desired direction. For example, again if a tap is opened at a higher height, the water when it falls and reaches the bottom has achieved the flow of turbulent nature.

The naming of the Reynolds number was done by Arnold Sommerfield in the year 1908 after its creator Osborne Reynolds, who studied and found the transition between the laminar and turbulent nature of the fluid. However, it is being said that the concept was introduced by George Stokes in the year 1851.

### Mathematical Formula

Mathematically the Reynolds number is defined as, the ratio of multiplication density and velocity of the fluid along with characteristics and length of the object over which flow is measured to the dynamic viscosity of the fluid.

It is represented in the form of formula as,

Reynolds Number (Re)=ρvl/μ

Where,

ρ= density of the particular fluid

v = velocity of the particular fluid

l = characteristics length of the fluid

μ= fluid’s dynamic viscosity

If the Reynolds number formula is used for any kind of flat or curved surface like a flat plane or an airfoil. The characteristic length is the length of the plate and for airfoil it is the chord of that airfoil.

And if the formula is being used for a rotating cylinder. Then the formula changes as,

Reynolds Number (Re)=ρvd/μ

Where,

d = Diameter of the Cylinder.

For flow inside of a pipe or cylinder, if the value of the Reynolds number is less than 2000, the flow is said to be laminar. And if it is greater than 4000, then the flow is said to be turbulent. And if it is between 2000 to 4000, the flow is said to be in transition mode (when it is converting from laminar to turbulent).

For flow when flowing outside anybody such as over a flat plate, if the value of Reynolds number is less than 5,00,000, the flow is said to be laminar, and if greater than 5,00,000 the flow is said to be turbulent.

**Also see:**

Trigonometry formula | Mean formula | Cube formula |

Percentage formula | Area of a square formula | Herons formula |

### Application

Used during determining the movement of bodies over water such as boats and other organisms or humans during swimming

Used during determining the aerodynamics of planes during their conceptual design period

Used during the design of pipelines in industries and factories

**Solved Example:**

**Find out whether the flow in a pipe having a diameter of 20 mm, fluid velocity of 2.5 m/s, 900 kg/m^3 as the relative density and the viscosity of the fluid as 0.4 Ns/m^2, the flow is laminar or turbulent.**

Given,

Fluid Velocity = 2.5 m/s

Relative Density = 900 Kg/m^3

Diameter of the Pipe = 20×10^{-3} m

Substituting the values in the formula,

Reynolds Number (Re)=ρvd/μ

Re=900×2.5×20×10^{-3}/0.4=112.5

Since the Reynolds number is less than 2000, the fluid is flowing in a laminar manner throughout the pipe.

**Important Maths Formulas:**

Simple interest formula | Mode formula | Fahrenheit to celsius formula |

Distance formula | Speed formula | Pythagorean theorem formula |

Standard deviation formula | Volume of a cylinder formula | Mean median mode formula |