# Concept behind Reynolds number

I would be glad if someone can explain to me the concepts and the applications of Reynolds number in real-life.

My handout has this explanation and I cannot understand it clearly...

Reynolds number denoted by $$N_R$$ is an experimentally deduced relationship which is used to determine whether the flow is laminar or turbulent in a flow tube. Mathematically,

$$N_R$$ = $$\frac{2pvr}{\eta}$$

where $$𝜌$$ is the density of the fluid, $$𝑣$$ is speed, $$𝑟$$ radius of the flow tube, and 𝜂 is the viscosity of the fluid. The flow is said to be laminar if $$N_R$$ is below 2000. When $$N_R$$ is between 2000-3000 the flow is said to be chaotic – the behavior is sensitive to small perturbations and very sensitive to predict. Above $$3000$$ the flow is characterized to be turbulent.

• youtube.com/watch?v=y7Hyc3MRKno Nov 8, 2020 at 20:18
• The pressure drop for fluid flowing through a pipe is determined quantitatively in terms of the Reynolds number. Nov 8, 2020 at 22:18
• The proper symbol for Reynolds number is ${\rm Re}$.
– Buzz
Nov 9, 2020 at 2:20

Dimensionless numbers in fluid dynamics are always a ratio of two quantities. The expression that you share is only a results of that expression. The Reynolds number is defined as the ratio between inertial forces and viscous forces.

Both forces can be approximated from the Navier-Stokes equations.

The inertial term is: $$\rho \vec{u} \cdot \nabla\vec{u}\sim \frac{\rho u^2}{D}$$

The viscous term: $$\mu \nabla^2\vec{u}\sim\mu\frac{u}{D^2}$$

Dividing these terms leads to the expression for the Reynolds number.

For low Reynolds number, the viscous forces are dominant. Therefore, the flow is laminar. For larger Reynolds number the inertia of the flow is a dominant factor, and viscous dissipation will not happen on the big scales. Therefore, the flow is turbulent.

There is no hard limit on when your flow is turbulent, and when it is laminar. That is why we would typically talk about regimes: laminar, transitional and turbulent.

One real-life application of the Reynolds number (no apostrophe as it is named after Osborne Reynolds) is in the design of small scale simulations of fluid dynamics scenarios, such as wind tunnel and water tank models. The Reynolds number is one of the dimensionless parameters that must be replicated in scale models in order to accurately reproduce the full scale flow pattern.

• Actually, the Reynolds number is often one of the parameters that is allowed to change in small-scale experiment in wind tunnels. In water tunnels one might be more lucky because the (kinematic) viscosity of water is much smaller, but for small scale models in wind tunnels the Reynolds number is often much lower and some kind of Reynolds-number independence is assumed. This is typical in wind engineering (building wind loading, pedestrian wind comfort), air pollution studies or models of very large vehicles (ships). ... Jul 16 at 8:10
• ... There were some experiments done in pressurized tunnels to get the smaller kinematic viscosity by increasing air density for wind farm models but it is very rare and expensive. Jul 16 at 8:13

I think your handout is written in quite a confusing way.

The main point of the Reynolds number is similarity. You can show that flows around a geometrically identical shape, but with different size, different flow speed, different viscosity will behave the same way if the Reynolds number is the same.

I like to write it in a different way, because I like short equations

$$\mathrm{Re} = \frac{U L}{\nu}$$ where $$U$$ is a characteristic flow speed, $$L$$ is a characteristic length scale (depends on the situation you are looking at, it can be the ball diameter, wire thickness, the height of the building, the boundary layer thickness, the distance from the leading edge...) and $$\nu = \eta / \rho$$ is the kinematic viscosity. Please not that it is more common to use $$\mu$$, rather than $$\eta$$ for the (dynamic) viscosity in fluid dynamics.

In your handout they chose $$L = 2r$$ where $$r$$ is likely a diameter of something. So the characteristic length scale is the diameter.

As Bernhard already wrote, the Reynolds number is the ratio of the inertial and viscous forces. The actual ratio differs in different points of the flow, but this is some characteristic representative value. The inertial forces mean that the flow is trying to continue to flow in the direction it is flowing and other forces have to act to change that.

The inertial term is nonlinear and is responsible for turbulence. If there is something that causes differences of the flow velocity in different location (shear or strain), the inertial term will cause creation of vortices that break up to smaller vortices and so on. Viscosity acts against this and with large enough viscosity, the differences in velocity can be sustained without turbulence.

They also mention some Reynolds number limits for the laminar and turbulent flow in your handout. Remember these are only valid for that particular type of flow with those particular definition of the characteristic length scale $$L$$ and the characteristic flow speed $$U$$. There are different limits for a flow around a ball, for a flow around a cylinder, for flows above surfaces, for pipe flows...

A turbulent flow is chaotic. The (deterministic) chaos is the main characteristic of turbulence. It means that only a minuscule change to the conditions will completely change the flow field configuration in the future and it is not possible to predict the future positions of each vortex (after some time). It has a direct connection to the predictability of atmosphere and why the weather forecast is only possible for about a week.