# Reynolds number: what is it for, how is it calculated, exercises

The **Reynolds number** ( *R _{e}* ) is a dimensionless numerical quantity that establishes the relationship between the inertial forces and viscosity forces of a fluid in motion. Inertial forces are determined by Newton’s second law and are responsible for the maximum acceleration of the fluid. Viscous forces are the forces that oppose fluid movement.

The Reynolds number applies to any type of fluid flow, such as flow in circular or non-circular ducts, in open channels, and flow around submerged bodies.

The value of the Reynolds number depends on the density, viscosity, fluid velocity and the dimensions of the current path. The behavior of a fluid, depending on the amount of energy that dissipates due to friction, will depend on whether the flow is laminar, turbulent or intermediate. For this reason, you need to find a way to determine the type of flow.

One way to determine this is by experimental methods, but they require very accurate measurements. Another way to determine the type of flow is by obtaining the Reynolds number.

In 1883, Osborne Reynolds discovered that if the value of this dimensionless number is known, the type of flow that characterizes any fluid conduction situation can be predicted.

__What is the Reynolds number for?__

__What is the Reynolds number for?__

The Reynolds number is used to determine the behavior of a fluid, that is, to determine whether the flow of a fluid is laminar or turbulent. Flow is laminar when viscous forces, which are opposite to the fluid’s movement, dominate and the fluid moves with a sufficiently small velocity and in a straight path.

Velocity of a fluid moving through a circular duct, for laminar flow (A) and turbulent flow (B and C). [By Olivier Cleynen (https://commons.wikimedia.org/wiki/File:Pipe_flow_velocity_distribution_laminar_turbulent.svg)]

The fluid with laminar flow behaves as if they were infinite layers that slide over each other, in an orderly manner, without mixing. In circular ducts, the laminar flow has a parabolic velocity profile, with maximum values in the center of the duct and minimum values in the layers close to the surface of the duct. The value of the Reynolds number in laminar flow is *R _{e} <2000* .

The flow is turbulent when inertial forces are dominant and the fluid travels with fluctuating changes in velocity and irregular trajectories. Turbulent flow is very unstable and exhibits momentum transfers between fluid particles.

When fluid circulates in a circular pipeline, with turbulent flow, the layers of fluid cross each other in eddies and their movement tends to be chaotic. The Reynolds number value for turbulent flow in a circular pipeline is *R _{e} > 4000.*

The transition between laminar flow and turbulent flow occurs for numerical Reynolds values between *2000* and *4000.*

__How is it calculated?__

__How is it calculated?__

The equation used to calculate the Reynolds number in a circular cross-section conduit is:

*R _{e} = ρVD / η*

*ρ* = fluid density ( *kg / m ^{3}* )

*V* = flow rate ( *m ^{3} / s* )

*D* = Linear dimension characteristic of the fluid path which, in the case of a circular duct, represents the diameter.

*η* = dynamic fluid viscosity ( *Pa.s* )

The relationship between viscosity and density is defined as kinematic viscosity *v* = *η / ρ,* and its unit is *m ^{2} / s* .

The equation of the Reynolds number as a function of kinematic viscosity is:

*R _{e} = DV / v*

In pipelines and channels with non-circular cross sections the characteristic dimension is known as hydraulic diameter *D _{H}* and represents a general dimension of the fluid path.

The generalized equation to calculate the Reynolds number in pipelines with non-circular cross sections is:

*R _{e} = ρV´D _{H}*

*/ η*

*V´ =* average flow = *V / A*

The hydraulic diameter *D _{H}* establishes the relationship between the area of

*the*cross section of the flow stream and the wetted perimeter

*P*.

_{M}*D _{H} = 4A / P _{M}*

The wetted perimeter *P _{M}* is the sum of the lengths of the walls of the duct, or channel, which are in contact with the fluid.

You can also calculate the Reynolds number of a fluid around an object. For example, a sphere immersed in a fluid that moves with velocity *V* . The sphere experiences a drag force *F _{R}* defined by the Stokes equation.

*F _{R} = 6πRVη*

*R* = radius of sphere

The Reynolds number of a sphere with velocity *V* submerged in a fluid is:

*R _{e} = ρV R *

*/ η*

R _{e} <1 when the flow is laminar and R _{e} > 1 when the flow is turbulent.

__solved exercises__

__solved exercises__

Below are three exercises to apply the Reynolds number: circular duct, rectangular duct, and sphere submerged in a fluid.

**Reynolds number in a circular duct**

Find the Reynolds propylene glycol number at *20°* C in a circular duct with a diameter of *0.5 cm* . The magnitude of the flow rate is *0.15 m ^{3} / s* . What is the flow type?

*D* = *0.5 cm* = *5.10 ^{-3} m* (characteristic dimension)

The density of the fluid is *ρ = 1,036 g / cm ^{3} = 1036 kg / m ^{3}*

The fluid viscosity is *η = **0.042 Pa · s = 0.042 kg / ms*

The flow is *V* = *0.15 m ^{3} / s*

The Reynolds number equation is used in a circular duct.

*R _{e} = *

*ρ*

*DV /*

*η*

*R _{and}* = (

*1036 kg / m*) /

^{3}x0,15m^{3}/ sx 5.10^{-3}m*(0.042 kg / ms) = 18.5*

The flux is laminar because the value of the Reynolds number is low in relation to the ratio *R _{e} <2000*

**Reynolds number in a rectangular duct**

Determine the type of flow of ethanol that flows with a velocity of 25 ml/min into a rectangular tube. The dimensions of the rectangular section are 0.5 cm and 0.8 cm.

Density *ρ =* 789 kg / m ^{3}

Dynamic viscosity *η = 1,074 mPa · s = 1,074.10 ^{-3} kg / ms*

First, the average flow is determined.

*V´* = *V / A*

*V = 25 ml / min = 4.16.10 ^{-7} m ^{3} / s*

The cross section is rectangular whose sides are 0.005m and 0.008m. The cross – section area is *A = 0.005 M x0.008m = 4.10 ^{-5} m ^{2}*

*V’* = *(4:16:10 ^{-7} m ^{3} / s) / (* 4:10

^{-5}m

^{2}) = 1.4 × 10

^{-2}m / s

The wet perimeter is the sum of the rectangle’s sides.

*P _{M}* =

*0,013m*

The hydraulic diameter is *D _{H} = 4A / P _{M}*

*D _{H}* =

*4 × 4.10*

^{-5}m^{2}/ 0.013 m*D _{H} = 1.23-10 ^{-2} m*

The Reynolds number is obtained from the equation *R _{e} = ρV´ D _{H}*

*/ η*

*R _{e}* = (789 kg / m

^{3}x1.04 × 10

^{-2}m / sx

*1.23.10*

^{-2}m) / 1,074.10^{-3}kg / ms*R _{e}* = 93974

The flow is turbulent because the Reynolds number is very large ( *R _{e} > 2000)*

**Reynolds the number of spheres submerged in a fluid**

A spherical particle, polystyrene latex, whose radius is *R **= 2000nm* is launched vertically into water with an initial velocity magnitude *V*= 10 m / s. Determine the Reynolds number of the particle submerged in water

Particle density *ρ = **1.04 g / cm ^{3} = 1040 kg / m ^{3}*

*R **= 2000nm = 0.000002m*

Water density *ρ _{ag}* =

*1000 kg / m*

^{3}Viscosity *η* = *0.001 kg / (m · s)*

The Reynolds number is obtained by the equation *R _{e} = ρV R *

*/ η*

*R _{e} = (1000 kg / m ^{3} x* 10 m / s

*x*0.000002m) /

*0.001 kg / (m · s)*

*R _{e} = 20*

The Reynolds number is 20. The flow is turbulent.

__applications__

__applications__

The Reynolds number plays an important role in fluid mechanics and heat transfer as it is one of the main parameters that characterize a fluid. Here are some of its applications.

1-It is used to simulate the movement of organisms that move on liquid surfaces, such as: bacteria suspended in water that swim through the fluid and produce random agitation.

2-It has practical applications in the flow of tubes and liquid circulation channels, confined flows, mainly in porous media.

3-In suspensions of solid particles immersed in fluid and emulsions.

4 – The Reynolds number is applied in tests in wind tunnels to study the aerodynamic properties of various surfaces, especially in the case of airplane flights.

5-It is used to model the movement of insects in the air.

6-The design of chemical reactors requires the use of Reynolds number to choose the flow model, taking into account the head losses, energy consumption and heat transmission area.

7-In the prediction of heat transfer of electronic components (1).

8-In the process of watering gardens and orchards, you need to know the flow of water that comes out of the pipes. To obtain this information, the hydraulic head loss is determined, which is related to the friction between the water and the pipe walls. The head loss is calculated once the Reynolds number is obtained.

**Applications in Biology**

In biology, the study of the movement of living organisms through water or in fluids with properties similar to water requires obtaining the Reynolds number, which will depend on the size of the organisms and the speed with which they move.

Bacteria and unicellular organisms have a very low Reynolds number ( *R _{e} << 1* ), hence the flow has a laminar velocity profile with a predominance of viscous forces.

Organisms with a size close to ants (up to 1 cm) have a Reynolds number of the order of 1, which corresponds to the transition regime in which the inertial forces acting on the organism are equally important as the viscous forces of the fluid.

In larger organisms, like people, the Reynolds number is very large ( *R _{e} >> 1* ).