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Reynolds Number Formula

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Last updated date: 19th Jul 2024
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What is Reynolds Number?

Reynolds Number is the ratio of the internal forces to the viscous forces in a fluid. This ratio depends on several factors, such as the internal motion due to different fluid velocities. The Reynolds Number is important in mechanics and is a dimensionless quantity, i.e. it had no units. The Reynolds Number has several applications, we shall discuss the applications as we progress further. Its primary application is the prediction of change from laminar to turbulent flow. 

A Sneak Peek into The History

The concept of Reynolds number, internal forces, viscous forces was introduced by George Stokes back in the year 1851. However, there was no solid theory behind it. The idea was further polished and built upon in the year 1908 by Arnold Sommerfeld. The name Reynold comes from Osborne Reynolds, the man who popularized the use of Reynolds Numbers in the year 1883. 

The Formula for Reynolds Number

The formula for Reynolds Number is as given below

\[ Re = \frac{(ρuL)}{μ} = \frac{(uL)}{ν}\]

In the formula given above, 

ρ Is rho or the density of the given fluid. 

u is the speed with which the fluid flows

μ is the dynamic viscosity of the fluid. 

L is the characteristic linear dimension. 

V is the kinematic velocity of the given fluid.

The Significance of Reynolds Number

So now that you've learned the formula of the Reynolds Number and who exactly formulated the Reynolds Number and in what year, you must be wondering what does this mere number signify. Well, the Reynolds Number signifies the nature of the flow of a fluid across a cross-section. On a wide basis, there are two types of flowing styles that a liquid can flow in. One is laminar flow and the other is turbulent flow. Reynolds number is used to determine the type of flow of a fluid. 

If Reynolds Number < 2300, then the liquid follows the laminar flow

If Reynolds number >2900, the liquid follows the turbulent flow

Reynolds Number-Its Application in Fluid Mechanics

The Reynolds number is used to study fluids as they flow. The Reynolds number determines whether a fluid flow is steady or unsteady. (laminar and turbulent) If a flow is laminar, fluids will move along smooth streamlines. If the flow is turbulent, these streamlines break up and the fluid will move irregularly.

It is used when modelling the movement of organisms swimming through water. It is used to predict the transition from laminar to turbulent flow and is used in the scaling of similar but different-sized flow situations, such as between an aircraft model in a wind tunnel and the full-size version.

Reynolds number, in fluid mechanics, is a criterion of whether the fluid (liquid or gas) flow is steady (streamlined, or laminar) or on average steady with small unsteady fluctuations (turbulent).

The transition between fluids from laminar to turbulent occurs very suddenly.

Reynolds number is given by

Reynolds Number = Inertial Force / Viscous Force

It is used when modeling the movement of organisms swimming through water. It is based on various factors like velocity, pressure, heat and temperature.

The Reynolds number is the ratio of inertial forces to viscous forces. The Reynolds number (Re) of a flowing fluid is computed by multiplying the fluid velocity by the pipe's internal diameter (to obtain the inertia force of the fluid) and then by dividing the result by the kinematic viscosity (viscous force per unit of  length).

Number Range

Actually, the transition between laminar and turbulent flow occurs not at a specific value of the Reynolds number but in a range usually beginning between 1,000 to 2,000 and extending upward to between 3,000 and 5,000.

The maximum range is between 2300 to 4000.The Turbulent flow occurs over a range of Reynolds numbers from approximately 2,300 to 4,000, regardless of the nature of the fluid or the dimensions of the pipe or the average velocity. All that matters is that this specific combination of the parameters, known as the Reynolds number, fall in the range indicated.

Types of Fluid Flow in Fluid Mechanics

The different types of fluid flow are:

  1. Steady and Unsteady Flow.

  2. Uniform and Non-Uniform Flow.

  3. Laminar and Turbulent Flow.

  4. Compressible and Incompressible Flow.

  5. Rotational and Irrotational Flow.

  6. One, Two and Three -dimensional Flow

FAQs on Reynolds Number Formula

1. In which areas the reynolds number formula is applicable?

The Reynolds number can be used to predict where this transition will take place. A vortex street around a cylinder. Reynolds number is applicable on fluids either be in liquid or gas form.For example if the candle is lit the Reynolds number will show the flame going from laminar to turbulent.It can also be applicable to wells, streams and lakes in liquid form. It is applicable to airplane,meteorological and climatic calculations.

2. On what factors does it depend?

R is the Reynolds number, which is unitless. It does depends on various factors like:

Flow Velocity:

If the  flow velocity increases, the Reynolds number also increases. 

Factor Dimension:

If the dimension of a factor increases the characteristic dimension of the Reynolds number increases.

Fluid Density:

If the density of fluid decreases the value of Reynolds number also decreases.

It also depends on pressure, temperature, characteristic length and the mainstream velocity for gas.

3. In which chapter of physics do we learn about reynolds number?

In mechanical properties of physics in 11 std students learn about Reynolds formula, its application, its range, units and its dimensions. In this chapter they also learn how Reynolds number is applicable in liquids and gasses and their flow from steady to unsteady in fluids. They also learn the application of Reynolds number in various fields with examples.

4. What is fluid dynamics?

Fluid mechanics is the branch of physics concerned with the mechanics of fluids and the forces on them. Fluid dynamics has a great range of applications in various fields of studies like various streams of engineering, like civil, mechanical or chemical. It is also applicable to the study of oceans, aircraft's meteorology and chemical industries.

5. What is the function of the reynolds number?

Reynolds number is indicative of the relative importance of inertial and viscous effects in fluid motion. Explanation: Reynolds number constitutes an important criterion of motion in fluids depending on factors like heat transfer and temperature. It also considers the density of the fluid. 

6. How is the reynolds number applicable in the pharmaceutical industry?

Purified water is the most used ingredient in pharmaceutical industries; it is used directly for the drug manufacturing process and also used for equipment's cleaning. If the Reynolds number is less than 2300 that means the flow of water is laminar. 

7. What is the type of flow when the reynolds number lies between 2300 and 2900?

This is the confusing range of Reynolds numbers. It is the range where it is difficult to determine the type of flow. It is the range where the type of flow is transiting from laminar to turbulent. In this range of Reynolds number, the flow could either be laminar or turbulent. It isn't easy to determine. It depends on factors like the surface through which the fluid is flowing. The flow could at times be laminar and at times be turbulent. The flow in this range is dynamic and could switch types from time to time. Hence, it's called intermittent flow. 

8. What does reynolds number depend on?

Reynolds number depends upon several factors. The factors on which the Reynolds Number depends are 

  • The density of the fluid: the density plays a key role in determining several physical factors of a fluid. Density is hence one of the primary factors that determine Reynolds Number. 

  • The speed with which the liquid is flowing in the cross-sectional area. 

  • Mu or the dynamic viscosity constant of the fluid. 

  • L, the length of the linear cross-section of the pipe. 

9. What are the applications of reynolds number?

The Reynolds Number does have several practical applications in the real-time world! The Reynolds Number's primary application is the analysis of the flow of a fluid in a cross-sectional area. It helps determine the velocity at which the flow transits from laminar to turbulent. The Reynolds Number is applied in determining the behaviours of different liquids. For instance, the movement of glycerine is predicted in a circular duct. The Reynolds Number is observed to be below 2300, hence we can conclude that the flow is laminar!