REYNOLDS NUMBER: WHAT IS IT FOR, HOW IS IT CALCULATED, EXERCISES - PHYSICAL - 2025

The Reynolds number (R _e) is a dimensionless numerical quantity that establishes the relationship between the inertial forces and the viscous 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 the movement of the fluid.

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

The value of the Reynolds number depends on the density, the viscosity, the speed of the fluid, and the dimensions of the current path. The behavior of a fluid as a function of the amount of energy that is dissipated, due to friction, will depend on whether the flow is laminar, turbulent or intermediate. For this reason it is necessary to find a way to determine the type of flow.

One way to determine it is by experimental methods but they require a lot of precision in the measurements. Another way to determine the type of flow is through obtaining the Reynolds number.

Water flow observed by Osborne Reynolds

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

What is the Reynolds number for?

The Reynolds number is used to determine the behavior of a fluid, that is, to determine if the flow of a fluid is laminar or turbulent. The flow is laminar when the viscous forces, which oppose the movement of the fluid, are those that dominate and the fluid moves with sufficiently small speed and in a rectilinear path.

Velocity of a fluid moving through a circular conduit, for laminar flow (A) and turbulent flow (B and C).

The fluid with laminar flow behaves as if it were infinite layers that slide one over the other, in an orderly way, without mixing. In circular ducts, laminar flow has a parabolic velocity profile, with maximum values ​​in the center of the duct and minimum values ​​in the layers near the duct surface. The Reynolds number value in laminar flow is R _e <2000.

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

When the fluid circulates in a circular conduit, with turbulent flow, the layers of fluid intersect each other forming eddies and their movement tends to be chaotic. The Reynolds number value for turbulent flow in a circular duct is R _e > 4000.

The transition between laminar flow and turbulent flow occurs for Reynolds number values ​​between 2000 and 4000.

How is it calculated?

The equation used to calculate the Reynolds number in a duct of circular cross section is:

In ducts and channels with non-circular cross sections the characteristic dimension is known as Hydraulic Diameter D _H and represents a generalized dimension of the fluid path.

The generalized equation for calculating the Reynolds number in conduits with non-circular cross sections is:

Hydraulic Diameter D _H establishes the relationship between the area A of the cross section of the current flow and the wetted perimeter P _M.

The wetted perimeter P _M is the sum of the lengths of the walls of the duct, or channel, that are in contact with the fluid.

You can also calculate the Reynolds number of a fluid that surrounds an object. For example, a sphere immersed in a fluid moving with speed V. The sphere experiences a drag force F _R defined by the Stokes equation.

R _e <1 when the flow is laminar and R _e > 1 when the flow is turbulent.

Solved exercises

Following are three Reynolds number application exercises: Circular conduit, Rectangular conduit, and Sphere immersed in a fluid.

Reynolds number in a circular duct

Calculate the Reynolds number of propylene glycol at 20 ° C in a circular duct with a diameter of 0.5 cm. The magnitude of the flow velocity is 0.15m ³ / s. What is the type of flow?

Viscosity of the fluid is η = 0.042 Pa s = 0.042 kg / ms

Flow velocity is V = 0.15m ³ / s

The Reynolds number equation is used in a circular duct.

The flow is laminar because the Reynolds number value is low with respect to the relation R _e <2000

Reynolds number in a rectangular duct

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

Density ρ = 789 kg / m ³

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

The average flow velocity is first determined.

The cross section is rectangular whose sides are 0.005m and 0.008m. The cross-sectional area is A = 0.005m x0.008m = 4.10 ^-5 m ²

The hydraulic diameter is D _H = 4A / P _M

The Reynolds number is obtained from the equation R _e = ρV´ D _H / η

Reynolds number of a sphere immersed in a fluid

A spherical latex polystyrene particle, whose radius is R = 2000nm, is launched vertically into the water with an initial velocity of magnitude V ₀ = 10 m / s. Determine the Reynolds number of the particle immersed in the water

Density of the particle ρ = 1.04 g / cm ³ = 1040 kg / m ³

Density of water 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

The Reynolds number is 20. The flow is turbulent.

Applications

The Reynolds number plays an important role in fluid mechanics and heat transfer because it is one of the main parameters that characterize a fluid. Some of its applications are mentioned below.

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 pipes and in liquid circulation channels, confined flows, in particular in porous media.

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

4-The Reynolds number is applied in wind tunnel tests to study the aerodynamic properties of various surfaces, especially in the case of aircraft flights.

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

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

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

8-In the process of watering the gardens and orchards in which it is necessary 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 that exists between the water and the pipe walls. The head loss is calculated once the Reynolds number is obtained.

Wind tunnel

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 are displace.

Bacteria and unicellular organisms have a very low Reynolds number (R _e << 1), consequently the flow has a laminar velocity profile with a predominance of viscous forces.

Organisms with a size close to ants (up to 1cm) 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 as important as the viscous forces of the fluid.

In larger organisms such as people the Reynolds number is very large (R _e >> 1).

References

1. Application of low-Reynolds number turbulent flow models to the prediction of electronic component heat transfer. Rodgers, P and Eveloy, V. NV: sn, 2004, IEEE, Vol. 1, pp. 495-503.
2. Mott, R L. Applied Fluid Mechanics. Berkeley, CA: Pearson Prentice Hall, 2006, Vol. I.
3. Collieu, AM and Powney, D J. The mechanical and thermal properties of materials. New YorK: Crane Russak, 1973.
4. Kay, JM and Nedderman, R M. An Introduction to Fluid Mechanics and Heat Transfer. New York: Cambridge Universitty Press, 1974.
5. Happel, J and Brenner, H. Mechanics of fluids and transport processes. Hingham, MA: MartinusS Nijhoff Publishers, 1983.