High School Physics

# Reynolds number & its formula, significance & sample numerical problem

The Reynolds number, Re is a dimensionless quantity and there is a ‘rule of thumb’ for flow in a pipe using the Reynolds number Re. When Re is less than 1000, flow can be taken to be laminar; when Re is greater than 2000, the flow will be turbulent. Reynolds number of a fluid flow depends on the radius of the pipe, speed of flow, density, and viscosity of the fluid. We will discuss this in detail in the following paragraphs.

## Lamina flow and Turbulent flow with examples

Laminar flow is a steady, predictable streamline flow where the stream tubes remain intact and the particles do not cross between streamlines.

Turbulent flow is unpredictable; it is characterized by the appearance of eddies and vortices in the fluid.

Steady (streamline) flow is observed at low fluid speeds.

As the speed increases, the flow between layers of fluid sliding past each other becomes unstable; particles from different streamlines and stream tubes begin to interact and mix. The flow is no longer laminar but is now turbulent.

To see an example of laminar flow turning into turbulent flow, watch the smoke rising from a piece of smoldering wood or paper. The smoke begins with an orderly flow but a few centimeters above the source of the smoke the flow becomes turbulent.

## Significance of Reynolds Number to detect the flow state

This transition between the two flow states and the transition speed are difficult to predict. There is a ‘rule of thumb’ for flow in a pipe using the Reynolds number Re.

When Reynolds number Re is less than 1000, flow can be taken to be laminar; when Re is greater than 2000, the flow will be turbulent. These values are different because there is a complex transition between the flow states—it is not possible to be precise about the nature of the flow in the transition.

## Reynolds Number formula

The Reynolds number, Re is a dimensionless quantity given by the formula as follows:
Re = v r ρ / η, where r is the radius of the pipe, v is the speed of flow, ρ is the density of the fluid, and η is its viscosity.

## Numerical problem – to find flow state using Reynolds number

Numerical Question: A syrup flows at a rate of 4.0m3 s -1 in a circular pipe of diameter 6.0cm.
The syrup has a density of 1300kgm-3 and a viscosity of 17Pa s.

Deduce whether the flow is laminar.

Solution :

flow rate = A v = Area x speed of flow

Speed of flow v = flow rate / area = 4 / [ π(0.03)2 ] = 1.4 x 103 m/s

Re = v r ρ / η = ( 1.4 x 103) (0.03) (1300) / 17 =3200

This is greater than 2000, so the flow will be turbulent.

## Related Posts

In this post, we have discussed the Reynolds number & its formula, significance & also solved a sample numerical problem. Now let us suggest a few interesting and related posts covering important topics from the fluid dynamics chapter. Here goes the list with relevant links.

Bernoulli’s Equation with derivation, explanation & examples

The Equation of Continuity with derivation & Streamlines of fluid flow

Bernoulli’s Principle with applications