What Are the Dimensions of Reynolds Number?

Key Takeaways

• Everything you need to know about the dimensions of Reynolds number.

• Why Reynolds number is important for fluid flow analysis.

• How to use dimensional analysis to calculate Reynolds number.

An example of fractal dimensions

There are many who point to nature as an example of the Law of Entropy. Or rather, left to her own accord, Mother Nature’s systems will tend toward more and more randomness and disorder. In some cases, this seems to be upheld. For example, as temperatures rise over time, ordered ice descends into disorder, with water molecules free to move in any manner. However, for some systems experiencing growth, such as a leaf on a fern tree, the opposite occurs as clear fractal dimensions emerge.

Fractal dimensions, which can be calculated by rearranging the basic scaling relation as given in the equation below, are in fact ratios of the level of complexity of a shape. Note that N is the number of fractals, S is the scaling factor, and D is the fractal dimension.

N = S-D ⇒ D = log(N)/log(S)

Although it is a specific quantity (has a value), the fractal dimension itself has no units and can be said to be dimensionless. The use of dimensionless quantities to define ratios between parameters of naturally occurring systems is not unique to fractals. In thermodynamics, one of the most used equations in fluid flow analysis is also a ratio and the dimensions of Reynolds number are also without units.

Are There Dimensions of Reynolds Number?

Prior to looking at the dimensions of Reynolds number, it is informative to examine what exactly is meant by the word dimension. The most common use of dimensionality is as a means of relating different geometric aspects of a shape. For example, length, width, and height define the three dimensions of a Cartesian system. However, dimension is also used to describe the unit of measurement for a quantity, which may be within a single dimension or direction.

Dimensions of Reynolds number from NASA

Reynolds number, as shown above, is used to determine the ratio of internal forces to viscous forces. While the forces, which do have units or dimensions, may be analyzed in multidimensions (1D, 2D, or 3D), the is a dimensionless quantity that is important for understanding the various stages of fluid flow.

Why Is Reynolds Number Important for Fluid Flow Analysis?

When designing mobile systems for operation on land, air, or sea, it is critical to understand the aerodynamics and/or fluid mechanics that control how the system operates. This is accomplished by being able to accurately model the fluid flow at and near the surface of the system (as for the vehicle below) as well as within a medium or conduit, such as a pipe. When modeling the fluid flow of systems, Reynolds number is helpful in defining the stage--laminar, transitional, or turbulent--of the fluid flow.

Aerodynamic forces of a vehicle

Reynolds number for fluid flow regions:

• Laminar
• ◆   Re < 2300
• Transient
• ◆   2300 < Re < 4000
• ➞ Turbulent
• ◆   Re > 4000

For pipes, the transition from laminar to turbulent flow is typically taken to occur at 2100. In order to determine the Reynolds number, data or information about the system must be known. And, depending on the system under study, the Reynolds number may be calculated using a number of different equations.

Reynolds number equations:

The equations above are not exhaustive. For example, the Reynolds number for a pipe can be calculated using the following equation as well.

This illustrates the flexibility in using the Reynolds number, as it can be calculated using different dimensions.

Using Dimensional Analysis to Calculate the Reynolds Number

As illustrated above, even though there are no dimensions of Reynolds number, dimensions play a significant role in calculating this important quantity.  For simple cases where fluid flow and system parameters are known, the basic steps of dimensional analysis may be sufficient. This involves applying unit conversion to ensure that all dimensions are canceled and the resulting Reynolds number is accurate.

For more complex problems, one of the formal dimensional analysis techniques, such as the Rayleigh or Buckingham 𝝿 methods that have well-defined algorithms or paradigms, may be applied. These are extremely helpful, as virtually any number of fluid flow parameters can be determined. However, the best solution for most systems is to forego manual calculations and apply a capable CFD tool for your system fluid flow analysis. For aerodynamics and hydraulics, an advanced means of performing system analysis, including calculating the parameters required to accurately define the dimensions of Reynolds number, is Cadence’s Omnis solver.