Reynolds number

The Reynolds number is a non-dimensional number defined as the ratio between the convective forces and the viscous forces of a fluid flow, and can be used to describe its nature, indicating whether it is laminar, transitional or fully turbulent.

What is the Reynolds Number?

The mathematical definition of the Reynolds number \( Re \) is

\begin{gather} Re=\frac{\rho U L}{\mu} \end{gather}

Where \(\rho\) is the fluid density, \(U\) the free stream velocity, \(L\) is a characteristic length and \(\mu\) is the dynamic molecular viscosity of the fluid.
The ratio \(\mu\) over \(\rho\) defines the kinematic molecular viscosity \(\nu\), hence the \(Re\) number can be expressed as

\begin{gather} Re=\frac{U L}{\nu} \end{gather}

Conversely to the density and viscosity, which are intrinsic properties of the fluid, the Reynolds number describes a property of the flow and indicates the relative importance of convective forces to viscous forces. When the viscous forces are dominant, for example in proximity of walls where the velocity of the flow approaches zero, the flow is laminar and it flows regularly. As the velocity of the flow increases, the convective forces start to overcome the viscosity that keeps the flow uniform and irregular chaotic patterns start to develop. This indicates the transition to a turbulent flow.

Flow past a 2D cylinder

The Reynolds number was named by the physicist  Arnold Sommerfeld in 1908 after the innovator Osborne Reynolds (1842–1912) who experimentally studied the transition of pipe flows from laminar to turbulent by varying the flow speed. The Reynolds number can therefore indicate if a flow is laminar, transitional or fully turbulent. A typical example of flow classification with the Reynolds number is the flow past a two-dimensional cylinder.

2D cylinder flow at different Re numbers
2D cylinder flow at different Re numbers

The unsteady vortex street is a phenomenon that can be seen in nature. In 1912, physicist Theodore von Kármán explained the mechanism of how long and spiralling cloud patterns are formed in the clouds. These so-called “von Kármán vortices” occur when winds flow around a large bluff obstacle, like for example an island raising above the sea level, creating a wake with alternating direction of rotation, which in turn form swirls in the clouds. The picture below shows cloud vortices swirling downwind of the Canary Islands and Madeira as captured by NASA’s Terra satellite on 20th May 2015. More interesting pictures are available on NASA’s website here.

Von Karman vortex street
Example of Von Karman vortex street in nature. Credits: Nasa

Dynamic similarity

In addition to flow classification, the Reynolds number can be used to estimate the transition from laminar to turbulent flow and can also be used to scale aerodynamics problems and obtain a similar flow at a different size or conditions. This concept is called dynamic similarity and is widely used in experimental aerodynamics to scale up or down a geometry while maintaining the same flow characteristics. Typical examples are scaled-down car models and aircraft models which are used in wind tunnel testing.

Passenger aeroplane model in a wind tunnel. Credits: NASA
Passenger aeroplane model in a wind tunnel. Credits: NASA

The role of the Reynolds number in CFD?

When solving fluid flows in a CFD simulation, the Reynolds number plays a significant role, as this describes the nature of the flow.
For low Reynolds number, where the flow is laminar, a relatively coarse mesh resolution is sufficient to resolve the flow. As the Reynolds number increases, so does the complexity of the flow structures to be solved. Therefore the mesh resolution necessary to correctly predict the flow in a CFD simulation increases significantly. Transitional and turbulent flows are prevalent in nature and in engineering applications, therefore the solution of turbulent flow is
one of the most important applications of CFD modelling, as discussed in our article about turbulence modelling in CFD.

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