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 geometricaly 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 lengthscale (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 thecharacteristic 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 inertaial 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 sustain 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 tyype of flow with those particular definition of the characteristic lengthscale $L$ 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.

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.