What is "reynolds number"

In fluid mechanics, the Reynolds number () is a dimensionless quantity that is used to help predict similar flow patterns in different fluid flow situations. The concept was introduced by George Gabriel Stokes in 1851, but the Reynolds number is named after Osborne Reynolds (1842–1912), who popularized its use in 1883.

The Reynolds number is defined as the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces for given flow conditions. Reynolds numbers frequently arise when performing scaling of fluid dynamics problems, and as such can be used to determine dynamic similitude between two different cases of fluid flow. They are also used to characterize different flow regimes within a similar fluid, such as laminar or turbulent flow:

• laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion;
• turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic eddies, vortices and other flow instabilities.

In practice, matching the Reynolds number is not on its own sufficient to guarantee similitude. Fluid flow is generally chaotic, and very small changes to shape and surface roughness can result in very different flows. Nevertheless, Reynolds numbers are a very important guide and are widely used.

Reynolds number interpretation has been extended into the area of arbitrary complex systems as well: financial flows, nonlinear networks, etc. In the latter case an artificial viscosity is reduced to nonlinear mechanism of energy distribution in complex network media. Reynolds number then represents a basic control parameter which expresses a balance between injected and dissipated energy flows for open boundary system. It has been shown that Reynolds critical regime separates two types of phase space motion: accelerator (attractor) and decelerator. High Reynolds number leads to a chaotic regime transition only in frame of strange attractor model.

The Reynolds number can be defined for several different situations where a fluid is in relative motion to a surface. These definitions generally include the fluid properties of density and viscosity, plus a velocity and a characteristic length or characteristic dimension. This dimension is a matter of convention – for example radius and diameter are equally valid to describe spheres or circles, but one is chosen by convention. For aircraft or ships, the length or width can be used. For flow in a pipe or a sphere moving in a fluid the internal diameter is generally used today. Other shapes such as rectangular pipes or non-spherical objects have an equivalent diameter defined. For fluids of variable density such as compressible gases or fluids of variable viscosity such as non-Newtonian fluids, special rules apply. The velocity may also be a matter of convention in some circumstances, notably stirred vessels. The Reynolds number is defined as

$$\mathrm{Re} = \dfrac{ \mbox{inertial forces} }{ \mbox{viscous forces} } = \dfrac{ \mbox{(mass)(acceleration)} }{ \mbox{(dynamic viscosity)(velocity/distance)(area)} } = \frac{(\rho L^3) (v^2 / L)}{\mu (v/L) L^2} = \frac{\rho v L}{\mu} = \frac{ v L}{\nu}$$

where:

• is the maximum velocity of the object relative to the fluid ( SI units: m/s)

• is a characteristic linear dimension, (travelled length of the fluid; hydraulic diameter when dealing with river systems) (m)

• is the dynamic viscosity of the fluid (Pa·s or N·s/m or kg/(m·s))

• (nu) is the kinematic viscosity (m/s)

• is the density of the fluid (kg/m).

Note that multiplying the Reynolds number by yields , which is the ratio of the inertial forces to the viscous forces. It could also be considered the ratio of the total momentum transfer to the molecular momentum transfer.