**Content**

**Reynolds number definition**

“The Reynolds number is the ratio of inertial forces to viscous forces.”

The Reynolds number is a dimensionless number used to study the fluid systems in various ways like the flow pattern of a fluid, the flow’s nature, and various fluid mechanics parameters. The Reynold’s number is also important in the study of heat transfer. There are much correlation developed, including Reynold’s number in fluid mechanics, tribology and heat transfer. The preparation of various medicines in pharmacy required Reynold’s number study.

It is actually a representation and comparison of inertia force and viscous force.

**Reynolds number equation**

The dimensionless Reynold’s number represents whether the flowing fluid would be laminar flow or turbulent flow, considering to some properties such as velocity, length, viscosity, and flow type. The Reynold’s number has been discussed as follow:

The Reynold’s number is generally termed as the inertia force ratio to viscous force and characterize the flow nature like laminar, turbulent etc. Let’s see by the equation as below,

[latex]Re= \frac{Inertia force}{viscous force}[/latex]

[latex]Inertia force =\rho A V^{2}[/latex]

[latex]Viscous force = \frac{\mu V A}{D}[/latex]

By putting the inertia force and viscous force expression in Reynold’s number expression, we get

[latex]Re = \frac{\\rho V D}{\mu }[/latex]

In above equation,

Re = Reynold’s number (Dimensionless number)

**?** = density of fluid (kg / m^{3})

V = velocity of flow ( m/ s )

D = Diameter of flow or pipe/ Characteristics length ( m )

μ = Viscosity of fluid (N *s /m^{2})

**Reynolds number units**

The Reynold’s number is dimensionless. There is no unit of Reynolds number.

**Reynolds number for laminar flow**

The identification of flow can be possible by knowing the Reynold’s number. The Reynold’s number of laminar flow is less than 2000. In an experiment, if you get a value of Reynold’s number less than 2000, then you can say that the flow is laminar.

**Reynolds number of water**

The equation of Reynold’s number is given as

[latex]Reynolds number= \frac{Density of fluid \cdot velocity of flow\cdot Diameter of flow/Length}{Viscosity of fluid}[/latex]

If we analyze the above equation, the Reynolds number’s value depends on the density of fluid, velocity of flow, the diameter of flow directly and inversely with the viscosity of the fluid. If the fluid is water, then the density and viscosity of water are the parameters that directly depend on water.

**Reynolds number for turbulent flow**

Generally, the Reynolds number experiment can predict the flow pattern. If the value of Reynold’s number is > 4000, then the flow is considered as turbulent nature.

**Drag Coefficient (Cd) vs Reynolds number (Re) in various objects**

**Reynolds number in a pipe**

If the fluid is flowing through the pipe, we want to calculate Reynold’s number of fluid flowing through a pipe. The other all parameters depends on the type of fluid, but the diameter is taken as pipe Hydraulics diameter D_{H }(For this, the flow should be properly coming out from the pipe)

[latex]Reynolds number= \frac{Density of fluid \cdot velocity of flow\cdot Hydraulic Diameter of flow/Length}{Viscosity of fluid}[/latex]

**Reynolds number of air**

As we have discussed in** Reynold number for water, **The Reynold number for air directly depends on air density and viscosity.

**Reynolds number range**

Reynold’s number is the criteria to know whether the flow is turbulent or laminar.

If we consider the flow is internal then,

If Re < (2000 to 2300) flow is considered laminar characteristics,

Re > 4000 represents turbulent flow

If Re’s value is in between (i.e. 2000 to 4000) represents transition flow.

**Reynolds number chart**

The moody chart is plotted between Reynolds number and friction factor for different roughness.

We can find the Darcy-Weisbach friction factor with Reynold number. There is an analytical correlation developed to find the friction factor.

**Reynolds number kinematic viscosity**

The kinematic viscosity is given as,

[latex]Kinematic viscosity = \frac{Viscosity of fluid}{Density of fluid}[/latex]

The Equation of Reynold’s number,

[latex]Reynolds number= \frac{Density of fluid \cdot velocity of flow\cdot Hydraulic Diameter of flow/Length}{Viscosity of fluid}[/latex]

The above equation is formed as below if write it in the form of kinematic viscosity,

[latex][Reynolds number= \frac{velocity of flow\cdot Hydraulic Diameter of flow/Length}{Kinematic Viscosity of fluid}[/latex]

[latex]Re =\frac{VD}{\nu }[/latex]

**Reynolds number cylinder**

If the fluid is flowing through the cylinder and we want to calculate Reynold number of fluid flowing through the cylinder. The other all parameters depends on the type of fluid, but the diameter is taken as Hydraulics diameter D_{H} (For this, the flow should be properly coming out from the cylinder)

**Reynolds number mass flow rate**

We then analyse the Reynold’s number equation if we want to see the relationship between the Reynold’s number and mass flow rate.

[latex]Re = \frac{\rho V D}{\mu }[/latex]

As we know from the continuity equation, the mass flow rate is expressed as below,

[latex]m =\rho \cdot A\cdot V[/latex]

By putting values of mass flow rate in the Reynolds number equation,

[latex]Re =\frac{m\cdot D}{A\cdot \mu }[/latex]

It can be clearly noted from the above expression that the Reynold’s number has a direct relation with the mass flow rate.

**Laminar vs turbulent flow Reynolds number |** **Reynolds number laminar vs turbulent**

Generally, in fluid mechanics, we are analyzing two types of flow. One is the laminar flow which occurs at low velocity, and another is the turbulent flow which generally occurs at high velocity. Its name describes the laminar flow as the fluid particles flow in the lamina (linear) throughout the flow. In turbulent flow, the fluid travels with random movement throughout the flow.

Let’s understand this important point in detail,

## Laminar Flow

In laminar flow, the adjacent layers of fluid particles do not intersect with each other and flows in parallel directions is known as laminar flow.

In the laminar flow, all fluid layers flow in a straight line.

- There possibility of occurrence of laminar flow when the fluid flowing with low velocity and the diameter of the pipe is small.
- The fluid flow with a Reynold’s number less than 2000 is considered laminar flow.
- The fluid flow is very linear. There is the intersection of adjacent layers of the fluid, and they flow parallel to each other and with the surface of the pipe.
- In laminar flow, the shear stress only depends on the fluid’s viscosity and independent of the density of the fluid.

## Turbulent Flow

The turbulent flow is opposite to the laminar flow. Here, In fluid flow, the adjacent layers of the flowing fluid intersect each other and do not flow parallel to each other, known as turbulent flow.

The adjacent fluid layers or fluid particles are not flowing in a straight line in a turbulent flow. They flow randomly in zigzag directions.

- The turbulent flow is possible if the velocity of the flowing fluid is high, and the diameter of the pipe is larger.
- The value of the Reynold’s number can identify the turbulent flow. If the value of Reynold’s number is more than 4000, then the flow is considered a turbulent flow.
- The flowing fluid does not flow unidirectional. There is a mixing or intersection of different fluid layers, and they do not flow in parallel directions to each other but intersecting each other.
- The shear stress depends on its density in a turbulent flow.

**Reynolds number for flat plate**

If we analyse the flow over a flat plate, then the Reynolds number is calculated by the flat plate’s characteristics length.

[latex]Re = \frac{\rho V L}{\mu }[/latex]

In the above equation, Diameter D is replaced by L, which is the characteristics length of flow over a flat plate.

**Reynolds number vs drag coefficient**

Suppose the Reynold’s number’s value is lesser than the inertia force. There is a higher viscous force getting dominance on inertia force.

If the fluid viscosity is higher, then the drag force is higher.

**Reynolds number of a sphere**

If you want to calculate it for this case, the formula is

[latex]Re = \frac{\rho V D}{\mu }[/latex]

Here, Diameter D is taken as Hydraulics diameter of a sphere in calculations like cylinder and pipe.

**What is Reynolds number?**

Reynold’s number is the ratio of inertia force to viscous force. Re indicates it. It is a dimensionless number.

[latex]Re= \frac{Inertia force}{viscous force}[/latex]

**Significance of Reynolds number** | **Physical significance of Reynolds number**

Reynold number is nothing but comparing of two forces. One is the inertia force, and the second is the viscous force. If we take both force ratio, it gives a dimensionless number known as Reynold number. This number helps to know flow characteristics and know which of the two forces impacts more on flow. The Reynold number is also important for flow pattern estimation.

Viscous force -> Higher -> Laminar flow -> Flow of oil

Inertia Force -> Higher -> Turbulent flow > Ocean waves

**Reynolds experiment**

Osborne Reynolds first performed the Reynolds experiment in 1883 and observe the water motion is laminar or turbulent in pattern.

This experiment is very famous in fluid mechanics. This experiment is widely used to determine and observe the three flow. In this experiment, the water flows through a glass tube or transparent pipe.

The dye is injected with water flow in a glass tube. You can notice the flow of dye inside the glass tube. If the dye has a different colour than water, it is clearly observable. If the dye is flowing inline or linear, then the flow is laminar. If it dye shows turbulence or not flowing in line, we can consider the turbulent flow. This experiment is simple and informative for students to learn about flow and Reynolds number.

**Critical Reynolds number**

The critical Reynolds number is the transition phase of laminar and turbulent flow region. When the flow is changing from laminar to turbulent, the Reynold’s number reading is considered a critical Reynold’s number. It is indicated as Re_{Cr.} For every geometry, this critical Reynold’s number will be different.

**Conclusion**

Reynolds number is important terms in the field of engineering and science. It is used in study of flow, heat transfer, pharma etc. We have elaborated this topic in detail because of its importance. We have included some practical questions and answers with this topic.

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